id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-018901 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Answer using clear logical steps: 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
$$U=[-15,15).$$
(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": "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-018902 | Topology: Sequences — Characterizing Closed Sets | 10 | 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
$$U=[4,18].$$
(a) Determine whether $U$ is open.
(b) Determ... | [
{
"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-018903 | Topology: Metric Spaces — Open Sets via Balls | 10 | Answer with a short justification: 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
$$U=[6,27).$$
(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": "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-018904 | Topology: Real Line — Boundary Behavior | 10 | 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
$$U=(-12,-5).$$
(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": "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-018905 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Warm-up: 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
$$U=[-5,-3].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\var... | [
{
"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... | 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-018906 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | 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
$$U=[-19,9).$$
(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. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-018907 | Topology: Metric Spaces — Open Sets via Balls | 10 | 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
$$U=[-3,1].$$
(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": "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-018908 | Topology: Real Line — Boundary Behavior | 10 | Warm-up: 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
$$U=(-5,-1].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\var... | [
{
"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-018909 | Topology: Metric Spaces — Open Sets via Balls | 10 | Checkpoint: 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 closed.
(c) Provide two different ... | [
{
"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-018910 | Topology: Real Line — Boundary Behavior | 10 | Provide a rigorous solution: 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
$$U=[-1,31).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justific... | [
{
"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-018911 | Topology: Metric Spaces — Open Sets via Balls | 10 | 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
$$U=[-8,3].$$
(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": "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-018912 | Topology: Metric Spaces — Open Sets via Balls | 10 | Work this out carefully: 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
$$U=[-12,14].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide t... | [
{
"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... | 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-018913 | Topology: Sequences — Characterizing Closed Sets | 10 | 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
$$U=[-8,31).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: o... | [
{
"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-018914 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Be explicit about assumptions: 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.
(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: $\\boxed{\\text{neither open nor closed}$.\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. |
math-018915 | Topology: Sequences — Characterizing Closed Sets | 10 | Solve and then verify: 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
$$U=(5,8].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two di... | [
{
"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-018916 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Solve and sanity-check: 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
$$U=(10,29).$$
(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": "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-018917 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Answer using clear logical 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
$$U=[9,18].$$
(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: $\\boxed{\\text{closed but not open}$.\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-018918 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Answer with a short justification: 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
$$U=[8,44).$$
(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: $\\boxed{\\text{neither open nor closed}$.\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. |
math-018919 | Topology: Sequences — Characterizing Closed Sets | 10 | Challenge: 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
$$U=[-5,7).$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-018920 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Challenge: 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
$$U=(-9,26).$$
(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": "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. |
math-018921 | Topology: Real Line — Boundary Behavior | 10 | 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
$$U=[-4,35).$$
(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": "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-018922 | Topology: Complements — Open/Closed Duality | 10 | Work carefully and justify each inference: 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
$$U=[-9,-3].$$
(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: $\\boxed{\\text{closed but not open}$.\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. |
math-018923 | Topology: Sequences — Characterizing Closed Sets | 10 | 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
$$U=(-6,7].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide... | [
{
"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-018924 | Topology: Complements — Open/Closed Duality | 10 | 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
$$U=[3,16].$$
(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": "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-018925 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | 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
$$U=(8,16).$$
(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": "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... | 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-018926 | Topology: Complements — Open/Closed Duality | 10 | 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
$$U=[-15,19).$$
(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": "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. |
math-018927 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Find the exact value: 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
$$U=(-2,24).$$
(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": "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... | 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-018928 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | State any required conditions first: 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
$$U=(0,5).$$
(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: $\\boxed{\\text{open but not closed}$.\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... | 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-018929 | Topology: Sequences — Characterizing Closed Sets | 10 | 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
$$U=[-8,6).$$
(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": "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-018930 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Keep the final answer in boxed form: 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
$$U=[0,37].$$
(a) Determine whether $U$ is open.
(b) Determine wh... | [
{
"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-018931 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | 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
$$U=[-16,-2).$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-018932 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | State any required conditions first: 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
$$U=[7,22).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different just... | [
{
"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-018933 | Topology: Complements — Open/Closed Duality | 10 | Challenge: 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
$$U=[10,16).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ju... | [
{
"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. |
math-018934 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | 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
$$U=[-7,10].$$
(a) Determine whether $U$ is open.
(b) Determine... | [
{
"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... | 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-018935 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Be explicit about assumptions: 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
$$U=[0,26).$$
(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": "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-018936 | Topology: Sequences — Characterizing Closed Sets | 10 | 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
$$U=(-14,-10).$$
(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": "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-018937 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Task: 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
$$U=[-7,-1].$$
(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": "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... | 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-018938 | Topology: Complements — Open/Closed Duality | 10 | Solve and include a self-check: 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
$$U=(8,40].$$
(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": "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-018939 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Provide both a computational and a conceptual explanation: 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
$$U=[5,19).$$
(a) 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": "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-018940 | Topology: Metric Spaces — Open Sets via Balls | 10 | 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
$$U=[4,21).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justific... | [
{
"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-018941 | Topology: Sequences — Characterizing Closed Sets | 10 | Problem: 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
$$U=(-4,35].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Pr... | [
{
"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. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-018942 | Topology: Real Line — Boundary Behavior | 10 | Where appropriate, name the theorem you use: 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
$$U=[-6,19].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two di... | [
{
"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... | 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-018943 | Topology: Metric Spaces — Open Sets via Balls | 10 | Answer with a short justification: 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
$$U=(-14,-3).$$
(a) Determine whether $U$ is open.
(b) Determine wh... | [
{
"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-018944 | Topology: Real Line — Boundary Behavior | 10 | Be explicit about assumptions: 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
$$U=(4,7].$$
(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": "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-018945 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | 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
$$U=[9,12].$$
(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: $\\boxed{\\text{closed but not open}$.\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-018946 | Topology: Sequences — Characterizing Closed Sets | 10 | Carefully track domains: 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 closed.
(c) Provide tw... | [
{
"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... | 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-018947 | Topology: Metric Spaces — Open Sets via Balls | 10 | 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
$$U=(-11,2].$$
(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... | 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-018948 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Work carefully and justify each inference: 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
$$U=[8,22).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differen... | [
{
"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... | 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-018949 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Explain each transformation: 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
$$U=[-19,-15].$$
(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": "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-018950 | Topology: Real Line — Boundary Behavior | 10 | Warm-up: 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
$$U=[-14,7].$$
(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": "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. |
math-018951 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Give reasoning, not just computation: 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
$$U=[-4,32).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different... | [
{
"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-018952 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Give a theorem-based 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
$$U=[-1,15).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"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-018953 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | 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
$$U=[10,15).$$
(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... | 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-018954 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Warm-up: 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
$$U=(-5,19).$$
(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... | 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-018955 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Give reasoning, not just computation: 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
$$U=[-20,-7).$$
(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": "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-018956 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Be explicit about assumptions: 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
$$U=(7,26].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificat... | [
{
"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. |
math-018957 | Topology: Real Line — Boundary Behavior | 10 | Solve and then verify: 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
$$U=(10,32].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: o... | [
{
"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-018958 | Topology: Sequences — Characterizing Closed Sets | 10 | Challenge: 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
$$U=(1,14).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different jus... | [
{
"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-018959 | Topology: Real Line — Boundary Behavior | 10 | Prompt: 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
$$U=(-14,7).$$
(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... | 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-018960 | Topology: Metric Spaces — Open Sets via Balls | 10 | Give reasoning, not just computation: 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
$$U=[-20,10].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differen... | [
{
"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-018961 | Topology: Metric Spaces — Open Sets via Balls | 10 | Solve and justify each 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
$$U=(4,7).$$
(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": "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-018962 | Topology: Metric Spaces — Open Sets via Balls | 10 | 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
$$U=(-20,13].$$
(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": "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-018963 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Explain what is being counted/optimized: 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
$$U=[8,43].$$
(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: $\\boxed{\\text{closed but not open}$.\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-018964 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Solve with 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
$$U=(-11,16].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide t... | [
{
"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-018965 | Topology: Metric Spaces — Open Sets via Balls | 10 | Make each step logically reversible (or explain if not): 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
$$U=(-3,17].$$
(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... | 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-018966 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Task: 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
$$U=(-19,20).$$
(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": "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-018967 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Answer with a short justification: 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
$$U=[8,19].$$
(a) Determine whether $U$ is open.
(b) Determine whet... | [
{
"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. |
math-018968 | Topology: Complements — Open/Closed Duality | 10 | Work carefully and justify each inference: 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
$$U=(-20,-10].$$
(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... | 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-018969 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | 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
$$U=(-16,-3].$$
(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": "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. |
math-018970 | Topology: Metric Spaces — Open Sets via Balls | 10 | 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
$$U=[-12,6].$$
(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": "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. |
math-018971 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | 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
$$U=[-14,-7].$$
(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": "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-018972 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Proceed methodically: 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
$$U=[-13,24].$$
(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": "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-018973 | Topology: Sequences — Characterizing Closed Sets | 10 | Warm-up: 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
$$U=[-11,1).$$
(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": "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-018974 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | 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
$$U=(9,24).$$
(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": "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... | 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-018975 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | 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
$$U=(-10,8].$$
(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": "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-018976 | Topology: Sequences — Characterizing Closed Sets | 10 | Solve and sanity-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
$$U=[6,24).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two ... | [
{
"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-018977 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | 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
$$U=(-19,-16).$$
(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... | 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-018978 | Topology: Sequences — Characterizing Closed Sets | 10 | Indicate where a theorem is used: 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
$$U=[0,35].$$
(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": "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... | 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-018979 | Topology: Complements — Open/Closed Duality | 10 | Question: 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
$$U=(-19,-15].$$
(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": "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-018980 | Topology: Complements — Open/Closed Duality | 10 | 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
$$U=[3,10).$$
(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": "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. |
math-018981 | Topology: Complements — Open/Closed Duality | 10 | State any required conditions first: 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
$$U=(-5,22).$$
(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: $\\boxed{\\text{open but not closed}$.\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... | 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-018982 | Topology: Real Line — Boundary Behavior | 10 | Give reasoning, not just computation: 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
$$U=(-14,15).$$
(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": "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... | 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-018983 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Task: 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
$$U=[9,28).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\varepsi... | [
{
"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-018984 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Explain why your operations are valid: 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
$$U=(4,31).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different... | [
{
"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-018985 | Topology: Sequences — Characterizing Closed Sets | 10 | Give an answer and a quick verification: 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
$$U=[2,4].$$
(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": "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... | 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-018986 | Topology: Real Line — Boundary Behavior | 10 | Warm-up: 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
$$U=(2,42).$$
(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": "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... | 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-018987 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | 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
$$U=[-3,3].$$
(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": "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-018988 | Topology: Metric Spaces — Open Sets via Balls | 10 | Write the solution set clearly: 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
$$U=[9,37].$$
(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: $\\boxed{\\text{closed but not open}$.\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-018989 | Topology: Sequences — Characterizing Closed Sets | 10 | Solve and justify each 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
$$U=(7,9).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificat... | [
{
"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. |
math-018990 | Topology: Sequences — Characterizing Closed Sets | 10 | Proceed methodically: 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
$$U=(-8,18].$$
(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. |
math-018991 | Topology: Sequences — Characterizing Closed Sets | 10 | 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
$$U=(4,44].$$
(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": "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-018992 | Topology: Sequences — Characterizing Closed Sets | 10 | Give an answer and a quick verification: 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
$$U=(-3,19].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different... | [
{
"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... | 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-018993 | Topology: Metric Spaces — Open Sets via Balls | 10 | Problem: 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
$$U=(7,30).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\vare... | [
{
"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-018994 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Give reasoning, not just computation: 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
$$U=[-9,7].$$
(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": "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... | 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-018995 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | 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
$$U=[-12,13).$$
(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": "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. |
math-018996 | Topology: Real Line — Boundary Behavior | 10 | 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
$$U=[-2,37).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"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-018997 | Topology: Metric Spaces — Open Sets via Balls | 10 | Work carefully and justify each inference: 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
$$U=(10,16).$$
(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": "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... | 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-018998 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Solve and sanity-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
$$U=[-17,-5).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide tw... | [
{
"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-018999 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | 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
$$U=[-13,10].$$
(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: $\\boxed{\\text{closed but not open}$.\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. (Here the result is $\boxed{\text{closed but not open}$.) |
math-019000 | Topology: Metric Spaces — Open Sets via Balls | 10 | 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
$$U=(-16,-11).$$
(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": "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... | 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 $\boxed{\text{open but not closed}$.) |
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