id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-018801 | Topology: Complements — Open/Closed Duality | 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
$$U=(-3,36].$$
(a) 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... | 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-018802 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Provide a rigorous solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-1,22).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificati... | [
{
"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-018803 | Topology: Sequences — Characterizing Closed Sets | 10 | Derive the result step-by-step: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(5,25].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-018804 | Topology: Metric Spaces — Open Sets via Balls | 10 | Explain why your operations are valid: 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
$$U=(-12,13).$$
(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": "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-018805 | Topology: Complements — Open/Closed Duality | 10 | Complete the analysis: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-8,24).$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018806 | 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
$$U=[5,16].$$
(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": "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. |
math-018807 | Topology: Sequences — Characterizing Closed Sets | 10 | Work this out carefully: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[8,22].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\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-018808 | Topology: Complements — Open/Closed Duality | 10 | Carefully track domains: 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
$$U=(-5,14).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificatio... | [
{
"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. |
math-018809 | 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
$$U=(-12,-8].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-018810 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Exercise: 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
$$U=[-11,12].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ju... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\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-018811 | Topology: Real Line — Boundary Behavior | 10 | Show all reasoning: 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
$$U=[0,21).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one u... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-018812 | Topology: Complements — Open/Closed Duality | 10 | Complete the analysis: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-14,-10].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificatio... | [
{
"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-018813 | Topology: Sequences — Characterizing Closed Sets | 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
$$U=[-13,2).$$
(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": "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-018814 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Complete the analysis: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,16).$$
(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... | 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-018815 | Topology: Sequences — Characterizing Closed Sets | 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
$$U=(-15,-12].$$
(a) Determine whether $U$ is open.
(b) Determi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-018816 | Topology: Metric Spaces — Open Sets via Balls | 10 | Track units/moduli carefully: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[9,26].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificati... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-018817 | Topology: Sequences — Characterizing Closed Sets | 10 | Question: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-19,18).$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-018818 | Topology: Metric Spaces — Open Sets via Balls | 10 | Determine the requested 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
$$U=[-20,16).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018819 | 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
$$U=[-8,23].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) P... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018820 | Real Analysis: Sets in R — Neighborhood Arguments | 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
$$U=(-19,-17).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ i... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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. |
math-018821 | Topology: Metric Spaces — Closed Sets via Limit Points | 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
$$U=(-7,30).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide tw... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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. |
math-018822 | Topology: Real Line — Boundary Behavior | 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
$$U=(3,39).$$
(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: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018823 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Checkpoint: 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
$$U=[7,15).$$
(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": "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-018824 | Real Analysis: Sets in R — Neighborhood Arguments | 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
$$U=[-6,5].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\va... | [
{
"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... | 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-018825 | Topology: Sequences — Characterizing Closed Sets | 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
$$U=[6,43).$$
(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... | 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-018826 | Topology: Complements — Open/Closed Duality | 10 | Explain what is being counted/optimized: 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
$$U=[-7,-4].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differ... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-018827 | Topology: Real Line — Boundary Behavior | 10 | Do not skip justification steps: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-20,-9].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justif... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018828 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Use two approaches if possible: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-18,10).$$
(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": "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-018829 | Topology: Sequences — Characterizing Closed Sets | 10 | Where appropriate, name the theorem you use: 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
$$U=(-10,14].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is 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: $\\boxed{\\text{neither open nor closed}$.\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-018830 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Find the exact value: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(1,5).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two dif... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-018831 | Topology: Complements — Open/Closed Duality | 10 | Exercise: 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
$$U=[8,15].$$
(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": "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-018832 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Derive the result step-by-step: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(4,34].$$
(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... | 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-018833 | Topology: Metric Spaces — Open Sets via Balls | 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
$$U=(-4,16).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ i... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\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-018834 | Topology: Sequences — Characterizing Closed Sets | 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
$$U=(-2,26].$$
(a) Determine whether $U$ is open.
(b) Determine whether $... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-018835 | Topology: Metric Spaces — Closed Sets via Limit Points | 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
$$U=[-8,17].$$
(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": "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-018836 | Real Analysis: Sets in R — Neighborhood Arguments | 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
$$U=[-16,0).$$
(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": "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-018837 | Topology: Metric Spaces — Open Sets via Balls | 10 | Do not skip justification steps: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,23).$$
(a) Determine whether $U$ is open.
(b) Determine whethe... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-018838 | Topology: Sequences — Characterizing Closed Sets | 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
$$U=(-8,16].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is clo... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018839 | Topology: Complements — Open/Closed Duality | 10 | Use two approaches if possible: 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
$$U=(-15,-1].$$
(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": "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-018840 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Keep the final answer in boxed form: 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
$$U=(-8,-1).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different jus... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018841 | Topology: Metric Spaces — Open Sets via Balls | 10 | State any required conditions first: 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
$$U=[-2,5].$$
(a) Determine whether $U$ is open.
(b) Determine wh... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-018842 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Solve and then verify: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-7,11].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018843 | Topology: Sequences — Characterizing Closed Sets | 10 | Prompt: 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
$$U=[9,48).$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018844 | Topology: Metric Spaces — Open Sets via Balls | 10 | Carefully track domains: 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
$$U=(-9,4).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justification... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\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. |
math-018845 | Topology: Sequences — Characterizing Closed Sets | 10 | Find the exact value: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(5,20].$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-018846 | Topology: Real Line — Boundary Behavior | 10 | Do not skip justification steps: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-14,6).$$
(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": "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-018847 | Topology: Complements — Open/Closed Duality | 10 | Exercise: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(1,6).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\vare... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-018848 | 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
$$U=(0,18].$$
(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": "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-018849 | Topology: Sequences — Characterizing Closed Sets | 10 | Provide a rigorous solution: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-19,15].$$
(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": "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-018850 | Topology: Real Line — Boundary Behavior | 10 | Where appropriate, name the theorem you use: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-19,20).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two diff... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-018851 | Topology: Complements — Open/Closed Duality | 10 | Exercise: 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
$$U=(-20,17).$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018852 | Topology: Metric Spaces — Open Sets via Balls | 10 | Solve and justify each step: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(9,35].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-018853 | Topology: Sequences — Characterizing Closed Sets | 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
$$U=[-11,-4].$$
(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": "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-018854 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Carefully track domains: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[3,31).$$
(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": "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-018855 | Topology: Sequences — Characterizing Closed Sets | 10 | Show all reasoning: 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
$$U=(5,23].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one u... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-018856 | Topology: Complements — Open/Closed Duality | 10 | Write the solution set clearly: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(3,40).$$
(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: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-018857 | Topology: Complements — Open/Closed Duality | 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
$$U=(-9,-3].$$
(a) Determine whether $U$ is open.
(b) Determine whether $... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018858 | Topology: Real Line — Boundary Behavior | 10 | Give a theorem-based solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-1,36).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018859 | Real Analysis: Sets in R — Neighborhood Arguments | 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
$$U=(8,13).$$
(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": "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-018860 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Solve and then verify: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-5,3).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications:... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018861 | Topology: Sequences — Characterizing Closed Sets | 10 | Compute the requested quantity: 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
$$U=[-13,25).$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-018862 | Topology: Complements — Open/Closed Duality | 10 | Use two approaches if possible: 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 justi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018863 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Provide a rigorous solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-14,-3].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificat... | [
{
"method_name": "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-018864 | Topology: Complements — Open/Closed Duality | 10 | Give a theorem-based solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(2,14].$$
(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": "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-018865 | Topology: Metric Spaces — Open Sets via Balls | 10 | Provide both a computational and a conceptual explanation: 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
$$U=[7,42].$$
(a) Determine whether $U$ is open.
(b) Determine whet... | [
{
"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. |
math-018866 | Topology: Real Line — Boundary Behavior | 10 | Prompt: 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
$$U=(-5,-1).$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018867 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Checkpoint: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-12,27].$$
(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": "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-018868 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Give a theorem-based 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
$$U=(-1,29].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justif... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018869 | 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
$$U=(8,30].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-018870 | Real Analysis: Sets in R — Neighborhood Arguments | 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
$$U=[-5,27].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two ... | [
{
"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-018871 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Work carefully and justify each inference: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-16,6).$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-018872 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Find the exact value: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-11,9].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: on... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018873 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Do not skip justification steps: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-3,27).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifi... | [
{
"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. (Here the result is $\boxed{\text{open but not closed}$.) |
math-018874 | Topology: Real Line — Boundary Behavior | 10 | Question: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(7,18].$$
(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": "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-018875 | Topology: Metric Spaces — Open Sets via Balls | 10 | Derive the result step-by-step: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(3,19].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018876 | Topology: Complements — Open/Closed Duality | 10 | Determine the requested value: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[5,27].$$
(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": "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. |
math-018877 | Topology: Sequences — Characterizing Closed Sets | 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
$$U=(-4,6].$$
(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... | 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-018878 | Topology: Metric Spaces — Closed Sets via Limit Points | 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
$$U=[-9,21).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\va... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-018879 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Work carefully and justify each inference: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,13).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is clo... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-018880 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Solve and sanity-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
$$U=[-17,-12).$$
(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-018881 | Topology: Metric Spaces — Open Sets via Balls | 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
$$U=(-20,14].$$
(a) Determine whether $U$ is open.
(b) Determin... | [
{
"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-018882 | Topology: Metric Spaces — Open Sets via Balls | 10 | Solve and then verify: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-8,29).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\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-018883 | Topology: Real Line — Boundary Behavior | 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
$$U=(-3,2).$$
(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": "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-018884 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Give reasoning, not just computation: 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
$$U=(-13,-6].$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-018885 | Topology: Sequences — Characterizing Closed Sets | 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
$$U=[-17,17).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-018886 | Topology: Metric Spaces — Open Sets via Balls | 10 | Where appropriate, name the theorem you use: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(1,12].$$
(a) Determine whether $U$ is open.
(b) Dete... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-018887 | Topology: Real Line — Boundary Behavior | 10 | Complete the analysis: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-10,18].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two... | [
{
"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-018888 | Topology: Complements — Open/Closed Duality | 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
$$U=[-15,-4].$$
(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": "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-018889 | Topology: Sequences — Characterizing Closed Sets | 10 | Answer using clear logical steps: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-12,28).$$
(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: $\\boxed{\\text{open but not closed}$.\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. |
math-018890 | Topology: Real Line — Boundary Behavior | 10 | Exercise: 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
$$U=[-15,8].$$
(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": "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-018891 | Topology: Sequences — Characterizing Closed Sets | 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
$$U=(4,28).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provid... | [
{
"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-018892 | Topology: Sequences — Characterizing Closed Sets | 10 | Use two approaches if possible: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-1,15).$$
(a) Determine whether $U$ is open.
(b) Determine whethe... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-018893 | 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
$$U=[-10,10].$$
(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: $\\boxed{\\text{closed but not open}$.\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. (Here the result is $\boxed{\text{closed but not open}$.) |
math-018894 | Topology: Real Line — Boundary Behavior | 10 | Use two approaches if possible: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-10,27).$$
(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: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018895 | Topology: Complements — Open/Closed Duality | 10 | Try to avoid pattern-matching; explain why: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-12,16].$$
(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": "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-018896 | Topology: Metric Spaces — Closed Sets via Limit Points | 10 | Do not skip justification steps: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-10,6].$$
(a) Determine whether $U$ is open.
(b) Determine wheth... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-018897 | Real Analysis: Sets in R — Neighborhood Arguments | 10 | Make each step logically reversible (or explain if not): 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
$$U=(10,50).$$
(a) Determine whether $U$ is open.
(b) Determine wheth... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-018898 | Topology: Sequences — Characterizing Closed Sets | 10 | Show all reasoning: 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
$$U=(-15,1).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one ... | [
{
"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-018899 | 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
$$U=(-2,11].$$
(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": "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-018900 | Topology: Metric Spaces — Open Sets via Balls | 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
$$U=[9,16].$$
(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": "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. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.