id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-014901 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Solve with verification: 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
$$U=(-14,1).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ i... | [
{
"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. (Here the result is $\boxed{\text{open but not closed}$.) |
math-014902 | Topology: Metric Spaces — Open Sets via Balls | 8 | Track units/moduli 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
$$U=[1,38].$$
(a) Determine whether $U$ is open.
(b) Determine whether $... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-014903 | Topology: Sequences — Characterizing Closed Sets | 8 | Give a theorem-based solution: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-19,7].$$
(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: $\\boxed{\\text{closed but not open}$.\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-014904 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Solve and include a self-check: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-7,20).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justific... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\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-014905 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | 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
$$U=(-12,26).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-014906 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Solve and include a self-check: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[3,40).$$
(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-014907 | Topology: Sequences — Characterizing Closed Sets | 8 | Solve and sanity-check: 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
$$U=[-3,28].$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-014908 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Solve with verification: 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
$$U=(3,39].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014909 | Topology: Metric Spaces — Open Sets via Balls | 8 | 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
$$U=(-15,0].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-014910 | Topology: Complements — Open/Closed Duality | 8 | 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
$$U=[-13,4].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: on... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-014911 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Give a fully justified 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
$$U=(6,9].$$
(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": "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-014912 | Topology: Sequences — Characterizing Closed Sets | 8 | Proceed methodically: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-11,29).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\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. |
math-014913 | Topology: Sequences — Characterizing Closed Sets | 8 | 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
$$U=[1,28).$$
(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": "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-014914 | Topology: Sequences — Characterizing Closed Sets | 8 | Where appropriate, name the theorem you use: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(10,27).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two di... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-014915 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Problem: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-16,17).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different jus... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-014916 | Topology: Sequences — Characterizing Closed Sets | 8 | 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
$$U=(-9,-5].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two d... | [
{
"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. |
math-014917 | Topology: Real Line — Boundary Behavior | 8 | 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
$$U=[-19,-3).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\v... | [
{
"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-014918 | Topology: Complements — Open/Closed Duality | 8 | 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
$$U=[-5,1).$$
(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": "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-014919 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Work this out carefully: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(1,23).$$
(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... | 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-014920 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Complete the analysis: 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
$$U=[-1,1].$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-014921 | Topology: Complements — Open/Closed Duality | 8 | 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
$$U=(-6,0].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justification... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-014922 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Proceed methodically: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[2,35).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is cl... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014923 | Topology: Complements — Open/Closed Duality | 8 | Start by stating any domain restrictions: 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
$$U=(-5,18).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is close... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014924 | Topology: Metric Spaces — Open Sets via Balls | 8 | 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
$$U=[-10,13).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ju... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014925 | Topology: Real Line — Boundary Behavior | 8 | 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
$$U=[10,22).$$
(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": "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-014926 | Topology: Complements — Open/Closed Duality | 8 | Solve and include a self-check: 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
$$U=(-1,34).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-014927 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Warm-up: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-10,18).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different jus... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-014928 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Give an answer and a quick verification: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-4,2).$$
(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": "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-014929 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Provide a rigorous solution: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-9,9).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014930 | Topology: Complements — Open/Closed Duality | 8 | Be explicit about assumptions: 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 open.
(b) Determine whether... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014931 | Topology: Real Line — Boundary Behavior | 8 | 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
$$U=(2,26).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\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-014932 | Topology: Metric Spaces — Open Sets via Balls | 8 | Write the solution set clearly: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-19,4].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justific... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014933 | Topology: Complements — Open/Closed Duality | 8 | 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
$$U=[-1,12).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justif... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-014934 | Topology: Real Line — Boundary Behavior | 8 | Solve and justify each step: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-19,18].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\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-014935 | Topology: Real Line — Boundary Behavior | 8 | State any required conditions first: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,13].$$
(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": "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-014936 | Topology: Complements — Open/Closed Duality | 8 | Compute the requested quantity: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[3,38].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-014937 | Topology: Real Line — Boundary Behavior | 8 | Track units/moduli carefully: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-9,27].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-014938 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Task: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(3,17].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justific... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-014939 | Topology: Complements — Open/Closed Duality | 8 | Explain what is being counted/optimized: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-12,22].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is close... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-014940 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Start by stating any domain restrictions: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-7,-3].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differen... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-014941 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | 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
$$U=[-14,26].$$
(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: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-014942 | Topology: Complements — Open/Closed Duality | 8 | Proceed methodically: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-3,1).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is cl... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\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. |
math-014943 | Topology: Metric Spaces — Open Sets via Balls | 8 | Problem: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(0,35).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\v... | [
{
"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-014944 | Topology: Real Line — Boundary Behavior | 8 | 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
$$U=[3,35].$$
(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... | 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-014945 | Topology: Sequences — Characterizing Closed Sets | 8 | Solve and sanity-check: 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
$$U=(1,14].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-014946 | Topology: Metric Spaces — Open Sets via Balls | 8 | Solve and include a self-check: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(1,17].$$
(a) Determine whether $U$ is open.
(b) Determine whether... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014947 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Compute the requested quantity: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(8,18).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014948 | Topology: Metric Spaces — Open Sets via Balls | 8 | 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
$$U=[8,38).$$
(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... | 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-014949 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Answer using clear logical steps: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-13,-4].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-014950 | Topology: Complements — Open/Closed Duality | 8 | Work carefully and justify each inference: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-15,21).$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014951 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Compute the requested quantity: 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
$$U=[7,22].$$
(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: $\\boxed{\\text{closed but not open}$.\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-014952 | Topology: Metric Spaces — Open Sets via Balls | 8 | Task: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-19,-4).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\varep... | [
{
"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-014953 | Topology: Sequences — Characterizing Closed Sets | 8 | Determine the requested value: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[2,40).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014954 | Topology: Sequences — Characterizing Closed Sets | 8 | 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
$$U=(-16,9).$$
(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": "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-014955 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Be explicit about assumptions: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-2,16].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-014956 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Task: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(10,15).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-014957 | Topology: Sequences — Characterizing Closed Sets | 8 | Start by stating any domain restrictions: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,29].$$
(a) Determine whether $U$ is open.
(b) Determi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-014958 | Topology: Metric Spaces — Open Sets via Balls | 8 | 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
$$U=(-14,10].$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-014959 | Topology: Real Line — Boundary Behavior | 8 | Make each step logically reversible (or explain if not): 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
$$U=(-9,24].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Pr... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014960 | Topology: Complements — Open/Closed Duality | 8 | 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
$$U=[3,15).$$
(a) Determine whether $U$ is open.
(b) Determine ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-014961 | Topology: Metric Spaces — Open Sets via Balls | 8 | Determine the requested value: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,10).$$
(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... | 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-014962 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | 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
$$U=(-20,-2].$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-014963 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Solve and include a self-check: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(4,38).$$
(a) Determine whether $U$ is open.
(b) Determine whether... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014964 | Topology: Real Line — Boundary Behavior | 8 | Start by stating any domain restrictions: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,11].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two diff... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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... | 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-014965 | Topology: Metric Spaces — Open Sets via Balls | 8 | Explain what is being counted/optimized: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-2,21).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014966 | Topology: Sequences — Characterizing Closed Sets | 8 | 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
$$U=(-10,8).$$
(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": "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-014967 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Warm-up: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-7,30].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different just... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014968 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | 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
$$U=(-13,6].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is clo... | [
{
"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. |
math-014969 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Start by stating any domain restrictions: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,24).$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-014970 | Topology: Metric Spaces — Open Sets via Balls | 8 | Checkpoint: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-4,27).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different j... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-014971 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | 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
$$U=[-3,34).$$
(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. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-014972 | Topology: Complements — Open/Closed Duality | 8 | Provide both a computational and a conceptual explanation: 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) Provi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014973 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | 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
$$U=[-19,5].$$
(a) Determine whether $U$ is open.
(b) Determine... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014974 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | 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
$$U=(-11,28].$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014975 | Topology: Metric Spaces — Open Sets via Balls | 8 | Complete the analysis: 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
$$U=(1,41].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is c... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-014976 | Topology: Sequences — Characterizing Closed Sets | 8 | Derive the result step-by-step: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[8,11).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justif... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014977 | Topology: Complements — Open/Closed Duality | 8 | 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
$$U=[-13,-7).$$
(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": "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-014978 | Topology: Complements — Open/Closed Duality | 8 | Track units/moduli carefully: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(2,5).$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-014979 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Warm-up: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-14,-5].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\va... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-014980 | Topology: Real Line — Boundary Behavior | 8 | 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
$$U=[-20,6).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two diffe... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-014981 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | 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
$$U=[-15,-3).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-014982 | Topology: Complements — Open/Closed Duality | 8 | Solve (and briefly cross-validate): Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(9,23].$$
(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-014983 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Give an answer and a quick verification: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-17,7).$$
(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... | 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-014984 | Topology: Metric Spaces — Open Sets via Balls | 8 | 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
$$U=[-5,18].$$
(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: $\\boxed{\\text{closed but not open}$.\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-014985 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | 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
$$U=[-20,1].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-014986 | Topology: Real Line — Boundary Behavior | 8 | 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
$$U=(-7,7].$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014987 | Topology: Sequences — Characterizing Closed Sets | 8 | Give a theorem-based solution: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[1,29).$$
(a) Determine whether $U$ is open.
(b) Determine whether ... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014988 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | Find the exact value: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-9,-2).$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-014989 | Real Analysis: Sets in R — Neighborhood Arguments | 8 | Task: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-19,3).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\vareps... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014990 | Topology: Metric Spaces — Open Sets via Balls | 8 | Show all reasoning: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-4,13).$$
(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": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014991 | Topology: Complements — Open/Closed Duality | 8 | 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
$$U=(-19,10].$$
(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": "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-014992 | Topology: Complements — Open/Closed Duality | 8 | 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
$$U=(-5,0].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Pr... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-014993 | Topology: Metric Spaces — Open Sets via Balls | 8 | Question: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-8,21).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-014994 | Topology: Sequences — Characterizing Closed Sets | 8 | Do not skip justification steps: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[5,25).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-014995 | Topology: Real Line — Boundary Behavior | 8 | 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
$$U=(-14,-1).$$
(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: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-014996 | Topology: Sequences — Characterizing Closed Sets | 8 | 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
$$U=[-3,25].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two diffe... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\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-014997 | Topology: Complements — Open/Closed Duality | 8 | Give a fully justified solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[3,7).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifica... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-014998 | Topology: Complements — Open/Closed Duality | 8 | Explain each transformation: 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
$$U=(-20,7).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificati... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-014999 | Topology: Metric Spaces — Closed Sets via Limit Points | 8 | 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
$$U=[-8,8).$$
(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": "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-015000 | Topology: Complements — Open/Closed Duality | 8 | Question: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-3,23).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\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 $\boxed{\text{open but not closed}$.) |
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