id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-016801 | Topology: Real Line — Boundary Behavior | 9 | Track quantifiers carefully: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-8,32).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justific... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016802 | Topology: Sequences — Characterizing Closed Sets | 9 | Find the exact value: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-8,28).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: on... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016803 | Topology: Complements — Open/Closed Duality | 9 | Show all reasoning: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(1,34).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is clos... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016804 | Topology: Complements — Open/Closed Duality | 9 | Work carefully and justify each inference: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,11].$$
(a) Determine whether $U$ is open.
(b) Determ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016805 | Topology: Complements — Open/Closed Duality | 9 | Try to avoid pattern-matching; explain why: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-15,14).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is cl... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016806 | Topology: Complements — Open/Closed Duality | 9 | Find the exact value: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-15,1).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two d... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016807 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Complete the analysis: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-8,4).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications:... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016808 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Start by stating any domain restrictions: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(3,42).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differ... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016809 | Topology: Metric Spaces — Open Sets via Balls | 9 | Explain what is being counted/optimized: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[10,27].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016810 | Topology: Metric Spaces — Open Sets via Balls | 9 | Compute the requested quantity: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-16,12).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016811 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Carefully track domains: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-6,20].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ i... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016812 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Explain each transformation: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-14,19).$$
(a) Determine whether $U$ is open.
(b) Determine whether ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016813 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Determine the requested value: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,5).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016814 | Topology: Complements — Open/Closed Duality | 9 | Answer using clear logical steps: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-7,12].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justif... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016815 | Topology: Complements — Open/Closed Duality | 9 | Solve (and briefly cross-validate): Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(9,29].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016816 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Work carefully and justify each inference: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[2,26).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is close... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016817 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Carefully track domains: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(4,14].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: ... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016818 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Show all reasoning: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(6,20].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two diff... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016819 | Topology: Metric Spaces — Open Sets via Balls | 9 | Where appropriate, name the theorem you use: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-6,22).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two diffe... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016820 | Topology: Sequences — Characterizing Closed Sets | 9 | Question: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-11,29].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ju... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016821 | Topology: Metric Spaces — Open Sets via Balls | 9 | Give an answer and a quick verification: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(9,19].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differe... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016822 | Topology: Complements — Open/Closed Duality | 9 | Solve and then verify: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-11,0].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016823 | Topology: Metric Spaces — Open Sets via Balls | 9 | Work this out carefully: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[4,19).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016824 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Indicate where a theorem is used: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-10,8).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) P... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016825 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Do not skip justification steps: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[3,37).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Pro... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016826 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Derive the result step-by-step: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(4,6].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016827 | Topology: Complements — Open/Closed Duality | 9 | Determine the requested value: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[7,30).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016828 | Topology: Real Line — Boundary Behavior | 9 | Track quantifiers carefully: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[3,43].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016829 | Topology: Complements — Open/Closed Duality | 9 | Question: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(0,32).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016830 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Solve (and briefly cross-validate): In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-13,19).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016831 | Topology: Real Line — Boundary Behavior | 9 | Solve and justify each step: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(3,23].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016832 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Question: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-11,7].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) P... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016833 | Topology: Sequences — Characterizing Closed Sets | 9 | Show all reasoning: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(0,9).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is close... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016834 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Derive the result step-by-step: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-16,21].$$
(a) Determine whether $U$ is open.
(b) Determine wheth... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016835 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Answer with a short justification: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-8,28].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ju... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016836 | Topology: Real Line — Boundary Behavior | 9 | Provide a rigorous solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[1,4].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justification... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016837 | Topology: Sequences — Characterizing Closed Sets | 9 | Write the solution set clearly: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(2,18].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016838 | Topology: Complements — Open/Closed Duality | 9 | Explain why your operations are valid: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-12,10].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016839 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Challenge: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[2,21].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\va... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016840 | Topology: Real Line — Boundary Behavior | 9 | Do not skip justification steps: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-5,26].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016841 | Topology: Real Line — Boundary Behavior | 9 | Warm-up: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-4,-1).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different just... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016842 | Topology: Sequences — Characterizing Closed Sets | 9 | Start by stating any domain restrictions: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-4,21].$$
(a) Determine whether $U$ is open.
(b) Determ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016843 | Topology: Sequences — Characterizing Closed Sets | 9 | Find the exact value: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(9,36].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is cl... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016844 | Topology: Metric Spaces — Open Sets via Balls | 9 | Work carefully and justify each inference: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,-10).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is cl... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016845 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Solve with verification: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-19,-2].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificati... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016846 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Start by stating any domain restrictions: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-10,30].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differe... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016847 | Topology: Complements — Open/Closed Duality | 9 | Provide a rigorous solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-14,-3).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificat... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016848 | Topology: Complements — Open/Closed Duality | 9 | Give a fully justified solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-10,26).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justif... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016849 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Problem: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-6,16].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016850 | Topology: Real Line — Boundary Behavior | 9 | Explain what is being counted/optimized: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(0,38].$$
(a) Determine whether $U$ is open.
(b) Determin... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016851 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Give a theorem-based solution: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-2,28].$$
(a) Determine whether $U$ is open.
(b) Determine whether... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016852 | Topology: Complements — Open/Closed Duality | 9 | Carefully track domains: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(3,12).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016853 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Indicate where a theorem is used: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-4,27].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) P... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016854 | Topology: Complements — Open/Closed Duality | 9 | Do not skip justification steps: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-6,0).$$
(a) Determine whether $U$ is open.
(b) Determine whethe... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016855 | Topology: Metric Spaces — Open Sets via Balls | 9 | Derive the result step-by-step: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,-8).$$
(a) Determine whether $U$ is open.
(b) Determine wheth... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016856 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Complete the analysis: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(6,14].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: on... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016857 | Topology: Complements — Open/Closed Duality | 9 | Keep the final answer in boxed form: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(6,41].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different j... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016858 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Solve (and briefly cross-validate): In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-19,1).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different j... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016859 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Track units/moduli carefully: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,-1).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016860 | Topology: Metric Spaces — Open Sets via Balls | 9 | Use two approaches if possible: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-7,14].$$
(a) Determine whether $U$ is open.
(b) Determine whethe... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016861 | Topology: Metric Spaces — Open Sets via Balls | 9 | Solve and include a self-check: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,6].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016862 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Try to avoid pattern-matching; explain why: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-6,22].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is clo... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016863 | Topology: Sequences — Characterizing Closed Sets | 9 | Start by stating any domain restrictions: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[3,19).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016864 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Write the solution set clearly: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-9,23].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Pro... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016865 | Topology: Metric Spaces — Open Sets via Balls | 9 | Give an answer and a quick verification: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-1,35).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016866 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Give a theorem-based solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-5,13).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016867 | Topology: Metric Spaces — Open Sets via Balls | 9 | Provide both a computational and a conceptual explanation: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-14,2).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016868 | Topology: Real Line — Boundary Behavior | 9 | Show all reasoning: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-9,2).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two diff... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016869 | Topology: Complements — Open/Closed Duality | 9 | Exercise: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-12,-6].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) ... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016870 | Topology: Sequences — Characterizing Closed Sets | 9 | Where appropriate, name the theorem you use: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-6,14).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two diffe... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016871 | Topology: Sequences — Characterizing Closed Sets | 9 | Solve (and briefly cross-validate): Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-15,7).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c)... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016872 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Solve and justify each step: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[3,40].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016873 | Topology: Metric Spaces — Open Sets via Balls | 9 | Solve and then verify: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-17,-2).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justification... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016874 | Topology: Sequences — Characterizing Closed Sets | 9 | Answer using clear logical steps: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-19,-7).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016875 | Topology: Real Line — Boundary Behavior | 9 | Track units/moduli carefully: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,31].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justific... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016876 | Topology: Real Line — Boundary Behavior | 9 | Problem: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-16,24].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different jus... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016877 | Topology: Real Line — Boundary Behavior | 9 | Solve and justify each step: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-5,15).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificati... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016878 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Solve and then verify: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-12,8].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016879 | Topology: Complements — Open/Closed Duality | 9 | Explain why your operations are valid: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[6,15].$$
(a) Determine whether $U$ is open.
(b) Determine ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016880 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Complete the analysis: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-18,-10).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide tw... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016881 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Where appropriate, name the theorem you use: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[4,29].$$
(a) Determine whether $U$ is open.
(b) Dete... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016882 | Topology: Metric Spaces — Open Sets via Balls | 9 | Provide a rigorous solution: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-5,34).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provid... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016883 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Use two approaches if possible: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-4,8).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016884 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Complete the analysis: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-12,-4).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justification... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016885 | Topology: Real Line — Boundary Behavior | 9 | Show all reasoning: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-17,-9].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016886 | Topology: Real Line — Boundary Behavior | 9 | Solve and include a self-check: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-18,-2].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Pr... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016887 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Do not skip justification steps: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[2,8).$$
(a) Determine whether $U$ is open.
(b) Determine whether... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016888 | Topology: Real Line — Boundary Behavior | 9 | Task: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-3,37].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-016889 | Topology: Real Line — Boundary Behavior | 9 | Task: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-17,-10).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\v... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016890 | Topology: Complements — Open/Closed Duality | 9 | Proceed methodically: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-1,4).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is cl... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016891 | Topology: Real Line — Boundary Behavior | 9 | Solve and then verify: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[2,21).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications:... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016892 | Topology: Sequences — Characterizing Closed Sets | 9 | Find the exact value: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-16,-5).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: o... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016893 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Solve and then verify: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(6,21).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is c... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016894 | Topology: Sequences — Characterizing Closed Sets | 9 | Give an answer and a quick verification: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(3,7].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016895 | Topology: Sequences — Characterizing Closed Sets | 9 | Question: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(6,15].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Pr... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016896 | Topology: Complements — Open/Closed Duality | 9 | Explain what is being counted/optimized: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-9,15).$$
(a) Determine whether $U$ is open.
(b) Determi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016897 | Topology: Complements — Open/Closed Duality | 9 | Derive the result step-by-step: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-1,18).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-016898 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Track quantifiers carefully: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-6,3).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-016899 | Topology: Complements — Open/Closed Duality | 9 | Work this out carefully: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(1,4].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-016900 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | State any required conditions first: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[9,38].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different j... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
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