Closed sets in metric spacesTopology - Open and closed sets in two different metric spacesContinuity in Metric SpacesCompact sets of metric spaces are closed?Closed sets and sequences in Metric spacesConfusion about compact subsets of metric spaces being closedTo show closedness of a subset in a metric spacesopen & closed sets in $mathbbR$ and $mathbbR^2$metric spaces proving the boundary of A is closedMetric Spaces - Open and Closed Subsetsmetric spaces: clarification on Open and Closed sets.
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Closed sets in metric spaces
Topology - Open and closed sets in two different metric spacesContinuity in Metric SpacesCompact sets of metric spaces are closed?Closed sets and sequences in Metric spacesConfusion about compact subsets of metric spaces being closedTo show closedness of a subset in a metric spacesopen & closed sets in $mathbbR$ and $mathbbR^2$metric spaces proving the boundary of A is closedMetric Spaces - Open and Closed Subsetsmetric spaces: clarification on Open and Closed sets.
$begingroup$
I have just proven any closed set in the plane (R^2) with the usual metric is the boundary of some subset of the plane.
I am now struggling to find a counterexample to show that this is not true for any metric space. Thanks for your time.
general-topology analysis metric-spaces
edited Mar 10 at 20:11
Andrés E. Caicedo
65.7k8160250
asked Mar 10 at 20:02
BrunoBruno
111
$endgroup$
add a comment |
$begingroup$
I have just proven any closed set in the plane (R^2) with the usual metric is the boundary of some subset of the plane.
I am now struggling to find a counterexample to show that this is not true for any metric space. Thanks for your time.
general-topology analysis metric-spaces
edited Mar 10 at 20:11
Andrés E. Caicedo
65.7k8160250
asked Mar 10 at 20:02
BrunoBruno
111
$endgroup$
$begingroup$
What have you tried?
$endgroup$
– James
Mar 10 at 20:03
$begingroup$
@TheSilverDoe Actually it is true.
$endgroup$
– Mark
Mar 10 at 20:10
$begingroup$
Oops sorry, I commented too quickly. My mistake !
$endgroup$
– TheSilverDoe
Mar 10 at 20:13
add a comment |
$begingroup$
I have just proven any closed set in the plane (R^2) with the usual metric is the boundary of some subset of the plane.
I am now struggling to find a counterexample to show that this is not true for any metric space. Thanks for your time.
general-topology analysis metric-spaces
edited Mar 10 at 20:11
Andrés E. Caicedo
65.7k8160250
asked Mar 10 at 20:02
BrunoBruno
111
$endgroup$
I have just proven any closed set in the plane (R^2) with the usual metric is the boundary of some subset of the plane.
I am now struggling to find a counterexample to show that this is not true for any metric space. Thanks for your time.
general-topology analysis metric-spaces
general-topology analysis metric-spaces
edited Mar 10 at 20:11
Andrés E. Caicedo
65.7k8160250
asked Mar 10 at 20:02
BrunoBruno
111
edited Mar 10 at 20:11
Andrés E. Caicedo
65.7k8160250
asked Mar 10 at 20:02
BrunoBruno
111
edited Mar 10 at 20:11
Andrés E. Caicedo
65.7k8160250
edited Mar 10 at 20:11
Andrés E. Caicedo
65.7k8160250
edited Mar 10 at 20:11
Andrés E. Caicedo
65.7k8160250
65.7k8160250
asked Mar 10 at 20:02
BrunoBruno
111
asked Mar 10 at 20:02
BrunoBruno
111
asked Mar 10 at 20:02
BrunoBruno
111
111
$begingroup$
What have you tried?
$endgroup$
– James
Mar 10 at 20:03
$begingroup$
@TheSilverDoe Actually it is true.
$endgroup$
– Mark
Mar 10 at 20:10
$begingroup$
Oops sorry, I commented too quickly. My mistake !
$endgroup$
– TheSilverDoe
Mar 10 at 20:13
add a comment |
$begingroup$
What have you tried?
$endgroup$
– James
Mar 10 at 20:03
$begingroup$
@TheSilverDoe Actually it is true.
$endgroup$
– Mark
Mar 10 at 20:10
$begingroup$
Oops sorry, I commented too quickly. My mistake !
$endgroup$
– TheSilverDoe
Mar 10 at 20:13
$begingroup$
What have you tried?
$endgroup$
– James
Mar 10 at 20:03
$begingroup$
What have you tried?
$endgroup$
– James
Mar 10 at 20:03
$begingroup$
@TheSilverDoe Actually it is true.
$endgroup$
– Mark
Mar 10 at 20:10
$begingroup$
@TheSilverDoe Actually it is true.
$endgroup$
– Mark
Mar 10 at 20:10
$begingroup$
Oops sorry, I commented too quickly. My mistake !
$endgroup$
– TheSilverDoe
Mar 10 at 20:13
$begingroup$
Oops sorry, I commented too quickly. My mistake !
$endgroup$
– TheSilverDoe
Mar 10 at 20:13
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Hint: Use the standard source of counterexamples: a discrete metric space.
answered Mar 10 at 20:05
José Carlos SantosJosé Carlos Santos
167k22132235
$endgroup$
$begingroup$
Okay got it thanks!
$endgroup$
– Bruno
Mar 10 at 20:43
$begingroup$
If my answer was useful, perhaps that you could mark it as the accepted one.
$endgroup$
– José Carlos Santos
Mar 10 at 21:40
add a comment |
$begingroup$
Consider the space $E = lbrace 0, 1 rbrace$, endowed with the induced metric.
The subset $lbrace 0 rbrace$ is closed, but is not the boundary of any subset.
answered Mar 10 at 20:21
TheSilverDoeTheSilverDoe
3,837112
$endgroup$
add a comment |
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: Use the standard source of counterexamples: a discrete metric space.
answered Mar 10 at 20:05
José Carlos SantosJosé Carlos Santos
167k22132235
$endgroup$
$begingroup$
Okay got it thanks!
$endgroup$
– Bruno
Mar 10 at 20:43
$begingroup$
If my answer was useful, perhaps that you could mark it as the accepted one.
$endgroup$
– José Carlos Santos
Mar 10 at 21:40
add a comment |
$begingroup$
Hint: Use the standard source of counterexamples: a discrete metric space.
answered Mar 10 at 20:05
José Carlos SantosJosé Carlos Santos
167k22132235
$endgroup$
$begingroup$
Okay got it thanks!
$endgroup$
– Bruno
Mar 10 at 20:43
$begingroup$
If my answer was useful, perhaps that you could mark it as the accepted one.
$endgroup$
– José Carlos Santos
Mar 10 at 21:40
add a comment |
$begingroup$
Hint: Use the standard source of counterexamples: a discrete metric space.
answered Mar 10 at 20:05
José Carlos SantosJosé Carlos Santos
167k22132235
$endgroup$
Hint: Use the standard source of counterexamples: a discrete metric space.
answered Mar 10 at 20:05
José Carlos SantosJosé Carlos Santos
167k22132235
answered Mar 10 at 20:05
José Carlos SantosJosé Carlos Santos
167k22132235
answered Mar 10 at 20:05
José Carlos SantosJosé Carlos Santos
167k22132235
answered Mar 10 at 20:05
José Carlos SantosJosé Carlos Santos
167k22132235
167k22132235
$begingroup$
Okay got it thanks!
$endgroup$
– Bruno
Mar 10 at 20:43
$begingroup$
If my answer was useful, perhaps that you could mark it as the accepted one.
$endgroup$
– José Carlos Santos
Mar 10 at 21:40
add a comment |
$begingroup$
Okay got it thanks!
$endgroup$
– Bruno
Mar 10 at 20:43
$begingroup$
If my answer was useful, perhaps that you could mark it as the accepted one.
$endgroup$
– José Carlos Santos
Mar 10 at 21:40
$begingroup$
Okay got it thanks!
$endgroup$
– Bruno
Mar 10 at 20:43
$begingroup$
Okay got it thanks!
$endgroup$
– Bruno
Mar 10 at 20:43
$begingroup$
If my answer was useful, perhaps that you could mark it as the accepted one.
$endgroup$
– José Carlos Santos
Mar 10 at 21:40
$begingroup$
If my answer was useful, perhaps that you could mark it as the accepted one.
$endgroup$
– José Carlos Santos
Mar 10 at 21:40
add a comment |
$begingroup$
Consider the space $E = lbrace 0, 1 rbrace$, endowed with the induced metric.
The subset $lbrace 0 rbrace$ is closed, but is not the boundary of any subset.
answered Mar 10 at 20:21
TheSilverDoeTheSilverDoe
3,837112
$endgroup$
add a comment |
$begingroup$
Consider the space $E = lbrace 0, 1 rbrace$, endowed with the induced metric.
The subset $lbrace 0 rbrace$ is closed, but is not the boundary of any subset.
answered Mar 10 at 20:21
TheSilverDoeTheSilverDoe
3,837112
$endgroup$
add a comment |
$begingroup$
Consider the space $E = lbrace 0, 1 rbrace$, endowed with the induced metric.
The subset $lbrace 0 rbrace$ is closed, but is not the boundary of any subset.
answered Mar 10 at 20:21
TheSilverDoeTheSilverDoe
3,837112
$endgroup$
Consider the space $E = lbrace 0, 1 rbrace$, endowed with the induced metric.
The subset $lbrace 0 rbrace$ is closed, but is not the boundary of any subset.
answered Mar 10 at 20:21
TheSilverDoeTheSilverDoe
3,837112
answered Mar 10 at 20:21
TheSilverDoeTheSilverDoe
3,837112
answered Mar 10 at 20:21
TheSilverDoeTheSilverDoe
3,837112
answered Mar 10 at 20:21
TheSilverDoeTheSilverDoe
3,837112
3,837112
add a comment |
add a comment |
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$begingroup$
What have you tried?
$endgroup$
– James
Mar 10 at 20:03
$begingroup$
@TheSilverDoe Actually it is true.
$endgroup$
– Mark
Mar 10 at 20:10
$begingroup$
Oops sorry, I commented too quickly. My mistake !
$endgroup$
– TheSilverDoe
Mar 10 at 20:13