id string | topic string | difficulty int64 | problem_statement string | solution_paths list | reconciliation dict | error_catalogue list | conceptual_takeaway string |
|---|---|---|---|---|---|---|---|
math-017001 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Explain why your operations are valid: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(9,45].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ju... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017002 | Topology: Complements — Open/Closed Duality | 9 | 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
$$U=(3,41].$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017003 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Explain why your operations are valid: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-8,14).$$
(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": "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-017004 | Topology: Real Line — Boundary Behavior | 9 | Keep the final answer in boxed form: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[4,26).$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017005 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Indicate where a theorem is used: 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
$$U=[8,37].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017006 | Topology: Complements — Open/Closed Duality | 9 | Write the solution set clearly: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-11,16).$$
(a) Determine whether $U$ is open.
(b) Determine wheth... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017007 | Topology: Real Line — Boundary Behavior | 9 | Solve and include a self-check: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(3,26].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017008 | Topology: Real Line — Boundary Behavior | 9 | Indicate where a theorem is used: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[8,34].$$
(a) Determine whether $U$ is open.
(b) Determine wheth... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017009 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Work this out carefully: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-7,33).$$
(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": "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-017010 | Topology: Real Line — Boundary Behavior | 9 | Proceed methodically: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-6,-1].$$
(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. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017011 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | 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
$$U=(9,20).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-017012 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Start by stating any domain restrictions: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-11,22).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differe... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-017013 | Topology: Sequences — Characterizing Closed Sets | 9 | 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
$$U=[4,27).$$
(a) Determine whether $U$ is open.
(b) Determine whether ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017014 | Topology: Metric Spaces — Open Sets via Balls | 9 | 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
$$U=[-17,-12].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ju... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017015 | Topology: Sequences — Characterizing Closed Sets | 9 | Problem: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,-14].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-017016 | Topology: Metric Spaces — Open Sets via Balls | 9 | Track units/moduli carefully: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(2,21).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificati... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-017017 | Topology: Sequences — Characterizing Closed Sets | 9 | Compute the requested quantity: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[3,33].$$
(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: $\\boxed{\\text{closed but not open}$.\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-017018 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | 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
$$U=[9,11).$$
(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": "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-017019 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Find the exact value: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[6,18).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is cl... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017020 | Topology: Complements — Open/Closed Duality | 9 | Solve (and briefly cross-validate): In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-14,18).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-017021 | Topology: Real Line — Boundary Behavior | 9 | Derive the result step-by-step: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-2,27].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Pro... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\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-017022 | Topology: Complements — Open/Closed Duality | 9 | Solve and then verify: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-13,-10].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificatio... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017023 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | State any required conditions first: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-14,20).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017024 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | 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
$$U=(-3,4).$$
(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: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-017025 | Topology: Metric Spaces — Open Sets via Balls | 9 | Give a fully justified solution: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(8,10].$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-017026 | Topology: Metric Spaces — Open Sets via Balls | 9 | Explain what is being counted/optimized: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-10,1].$$
(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... | 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-017027 | Topology: Sequences — Characterizing Closed Sets | 9 | Solve and sanity-check: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-16,18].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications:... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017028 | Topology: Complements — Open/Closed Duality | 9 | Explain what is being counted/optimized: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-13,9].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-017029 | Topology: Sequences — Characterizing Closed Sets | 9 | 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
$$U=[-2,10].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justif... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017030 | Topology: Real Line — Boundary Behavior | 9 | State any required conditions first: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-4,18].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-017031 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Keep the final answer in boxed form: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-1,2].$$
(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": "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-017032 | Topology: Complements — Open/Closed Duality | 9 | Be explicit about assumptions: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-7,5].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifi... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\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-017033 | Topology: Metric Spaces — Open Sets via Balls | 9 | Use two approaches if possible: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(0,12).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-017034 | Topology: Complements — Open/Closed Duality | 9 | Track quantifiers 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
$$U=(-11,-4].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificat... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017035 | Topology: Sequences — Characterizing Closed Sets | 9 | Do not skip justification steps: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-9,22].$$
(a) Determine whether $U$ is open.
(b) Determine wheth... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017036 | Topology: Sequences — Characterizing Closed Sets | 9 | Solve and then verify: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-19,-4].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justification... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-017037 | Topology: Complements — Open/Closed Duality | 9 | 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
$$U=[-20,19).$$
(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... | 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-017038 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Question: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-19,0].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\va... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-017039 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Give reasoning, not just computation: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(7,13).$$
(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": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017040 | Topology: Metric Spaces — Open Sets via Balls | 9 | 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
$$U=[-9,5].$$
(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: $\\boxed{\\text{closed but not open}$.\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-017041 | Topology: Real Line — Boundary Behavior | 9 | 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
$$U=(-15,2].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justif... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017042 | Topology: Real Line — Boundary Behavior | 9 | Write the solution set clearly: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(4,36).$$
(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. (Here the result is $\boxed{\text{open but not closed}$.) |
math-017043 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | 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
$$U=(0,17].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017044 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Solve and include a self-check: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-3,8).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-017045 | Topology: Real Line — Boundary Behavior | 9 | 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
$$U=[-10,5].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different just... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017046 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | 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
$$U=(-16,11).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different jus... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017047 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Provide both a computational and a conceptual explanation: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,-6).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c)... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017048 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Explain what is being counted/optimized: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-1,1).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differe... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017049 | Topology: Complements — Open/Closed Duality | 9 | Prompt: 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
$$U=[-4,25).$$
(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": "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-017050 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Solve and justify each step: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[7,27].$$
(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: $\\boxed{\\text{closed but not open}$.\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-017051 | Topology: Metric Spaces — Open Sets via Balls | 9 | Make each step logically reversible (or explain if not): Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-16,23).$$
(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: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017052 | Topology: Real Line — Boundary Behavior | 9 | Give a theorem-based solution: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[1,18].$$
(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": "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-017053 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Prompt: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[8,13].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-017054 | Topology: Complements — Open/Closed Duality | 9 | Question: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,15].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{closed but not open}$.\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-017055 | Topology: Sequences — Characterizing Closed Sets | 9 | Explain what is being counted/optimized: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-4,24].$$
(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": "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-017056 | Topology: Metric Spaces — Open Sets via Balls | 9 | Complete the analysis: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(7,39).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: on... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017057 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Start by stating any domain restrictions: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-13,6).$$
(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": "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-017058 | Topology: Sequences — Characterizing Closed Sets | 9 | Provide both a computational and a conceptual explanation: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(2,15).$$
(a) Determine whether $U$ is ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017059 | Topology: Sequences — Characterizing Closed Sets | 9 | Problem: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[2,7].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Prov... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017060 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Solve and then verify: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(1,31].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: on... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017061 | Topology: Real Line — Boundary Behavior | 9 | Derive the result step-by-step: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-18,14].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017062 | Topology: Real Line — Boundary Behavior | 9 | Where appropriate, name the theorem you use: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-12,1).$$
(a) Determine whether $U$ is open.
(b) Det... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017063 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Question: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(1,19].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different just... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017064 | Topology: Sequences — Characterizing Closed Sets | 9 | Solve and then verify: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-3,29].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: o... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robu... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-017065 | Topology: Metric Spaces — Open Sets via Balls | 9 | Challenge: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(2,27].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017066 | Topology: Sequences — Characterizing Closed Sets | 9 | Determine the requested value: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-9,-4).$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017067 | Topology: Metric Spaces — Open Sets via Balls | 9 | Find the exact value: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-14,10].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-017068 | Topology: Metric Spaces — Open Sets via Balls | 9 | Determine the requested value: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(2,34].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificat... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017069 | Topology: Complements — Open/Closed Duality | 9 | Question: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-1,16).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different jus... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-017070 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Make each step logically reversible (or explain if not): Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,38).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provid... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017071 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | 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
$$U=(10,46).$$
(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": "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-017072 | Topology: Real Line — Boundary Behavior | 9 | Indicate where a theorem is used: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(7,27].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different just... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017073 | Topology: Complements — Open/Closed Duality | 9 | Checkpoint: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-8,-6].$$
(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": "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-017074 | Topology: Real Line — Boundary Behavior | 9 | Solve with verification: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(7,34).$$
(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": "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-017075 | Topology: Sequences — Characterizing Closed Sets | 9 | Track quantifiers 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
$$U=[7,26].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificatio... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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-017076 | Topology: Sequences — Characterizing Closed Sets | 9 | 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
$$U=[-9,10).$$
(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-017077 | Topology: Sequences — Characterizing Closed Sets | 9 | Track quantifiers 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
$$U=(-1,9].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017078 | Topology: Sequences — Characterizing Closed Sets | 9 | 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
$$U=[-7,0].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different just... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "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. |
math-017079 | Topology: Real Line — Boundary Behavior | 9 | Carefully track domains: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(9,30).$$
(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: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robustness... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{open but not closed}$.) |
math-017080 | Topology: Metric Spaces — Open Sets via Balls | 9 | 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
$$U=[-16,-10].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justificati... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for t... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017081 | Topology: Real Line — Boundary Behavior | 9 | Answer with a short justification: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-20,-3).$$
(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: $\\boxed{\\text{neither open nor closed}$.\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-017082 | Topology: Metric Spaces — Open Sets via Balls | 9 | 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
$$U=(-15,25].$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017083 | Topology: Metric Spaces — Open Sets via Balls | 9 | Start by stating any domain restrictions: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[7,41).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017084 | Topology: Sequences — Characterizing Closed Sets | 9 | Work this out carefully: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(10,26).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide tw... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{open but not closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017085 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Make each step logically reversible (or explain if not): 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
$$U=[-2,0].$$
(a) Determine whether $U$ is op... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017086 | Topology: Sequences — Characterizing Closed Sets | 9 | Work carefully and justify each inference: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[10,13).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differe... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017087 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Exercise: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-15,-11].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: one using $\... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Both approaches agree after simplification. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"robust... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017088 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Complete the analysis: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(10,13].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: o... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017089 | Topology: Metric Spaces — Open Sets via Balls | 9 | Keep the final answer in boxed form: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-16,-13).$$
(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": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Remember: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017090 | Topology: Metric Spaces — Open Sets via Balls | 9 | Solve (and briefly cross-validate): Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(7,27).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justi... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{open but not closed}$.\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-017091 | Topology: Sequences — Characterizing Closed Sets | 9 | Keep the final answer in boxed form: Classify the set $U$ as open/closed/neither in $(\mathbb{R},|\cdot|)$, using **two different definitions** (balls and sequences/complements):
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[7,16].$$
(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": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Takeaway: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{closed but not open}$.) |
math-017092 | Topology: Complements — Open/Closed Duality | 9 | Keep the final answer in boxed form: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-7,-2].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification f... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017093 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Give an answer and a quick verification: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-5,5].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{closed but not open}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same s... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Key idea: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. |
math-017094 | Topology: Sequences — Characterizing Closed Sets | 9 | Provide both a computational and a conceptual explanation: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-8,25].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) ... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Consistency verification shows both paths yield the identical boxed result. Final answer: $\\boxed{\\text{neither open nor closed}$.\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-017095 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Give an answer and a quick 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
$$U=(6,24].$$
(a) Determine whether $U$ is open.
(b) Determin... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017096 | Topology: Metric Spaces — Open Sets via Balls | 9 | Give reasoning, not just computation: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-15,25).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differen... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017097 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Explain what is being counted/optimized: Decide openness and closedness. Give one proof via neighborhoods and one via limit points:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[-11,-2).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two differen... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "Cross-check: both derivations land on the same invariant quantity. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the sa... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | Core principle: In $\mathbb{R}$, openness means every point has a small interval around it still inside the set; closedness means the set contains its limit points. Checking boundary behavior is the key. (Here the result is $\boxed{\text{neither open nor closed}$.) |
math-017098 | Topology: Metric Spaces — Closed Sets via Limit Points | 9 | Do not skip justification steps: Topology check: determine whether $U$ is open and/or closed, and justify with balls + a convergent sequence argument:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=(-13,23].$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) P... | [
{
"method_name": "Complement / Sequence Criterion",
"approach": "Use that $U$ is closed iff its complement is open, or equivalently iff every convergent sequence in $U$ has its limit in $U$.",
"steps": [
"Step 1: Describe the complement $\\mathbb{R}\\setminus U$ explicitly.",
"Step 2: Check ... | {
"consistency_check": "The two methods are consistent and must coincide. Final answer: $\\boxed{\\text{neither open nor closed}$.\nThe open-ball definition and the complement/sequence definition are equivalent in metric spaces. Therefore both must yield the same open/closed classification for the same set $U$.",
"... | [
{
"error_description": "Assumed any interval is open because it 'contains points between its endpoints'.",
"why_plausible": "Everyday language about 'open interval' can leak into formal reasoning.",
"why_wrong": "Openness depends on neighborhoods around every point; included endpoints generally fail the... | 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-017099 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | Proceed methodically: In the usual metric on $\mathbb{R}$, analyze $U$ (open? closed?). Provide two justifications:
In the metric space $(\mathbb{R},d)$ with $d(x,y)=|x-y|$, consider the set
$$U=[0,2).$$
(a) Determine whether $U$ is open.
(b) Determine whether $U$ is closed.
(c) Provide two different justifications: o... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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-017100 | Real Analysis: Sets in R — Neighborhood Arguments | 9 | 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
$$U=(-7,21].$$
(a) Determine whether $U$ is open.
(b) Determine whethe... | [
{
"method_name": "Epsilon-Ball Definition",
"approach": "Use: $U$ is open iff for every $x\\in U$ there exists $r>0$ with $B(x,r)\\subseteq U$; $U$ is closed iff it contains all its limit points.",
"steps": [
"Step 1: Analyze openness by taking an arbitrary $x\\in U$ and estimating distance to the... | {
"consistency_check": "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 $\boxed{\text{neither open nor closed}$.) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.