Intersection of Compact Sets Is Not Compact
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What is an example of a topological space $X$ such that $C,Ksubseteq X$; $C$ is closed; $K$ is compact; and $Ccap K$ is not compact?
I know that $X$ can be neither Hausdorff nor finite.
I am interested in this question because I recently read the following definition (in a Rudin book):
If $left(X,tauright)$ is a topological space and $inftynotin X$, then $left(X_infty,tau_inftyright)$, where $X_infty=Xcupleftinftyright$ and every $Uintau_infty$ is such that $Uintau$ or $U^csubseteq X$ is compact, is a topological space.
I believe that this definition requires that $U^csubseteq X$ be compact and closed.
Edit: The first question was my attempt to show that if $U,Vintau_infty$ are such that $Uintau$ and $V^csubseteq X$ is compact, then $left(Ucup Vright)^c=U^ccap V^c$ is not compact. However, this holds, as the comments show. A correct counter-example would be to show that if $U,Vintau_infty$ are such that $U^c,V^csubseteq X$ are compact, then $left(Ucup Vright)^c=U^ccap V^c$ is not compact, as Rob Arthan shows in his answer.
general-topology
asked Aug 9 at 21:39
Cleric
3,06632463
 |Â
show 2 more comments
up vote
3
down vote
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What is an example of a topological space $X$ such that $C,Ksubseteq X$; $C$ is closed; $K$ is compact; and $Ccap K$ is not compact?
I know that $X$ can be neither Hausdorff nor finite.
I am interested in this question because I recently read the following definition (in a Rudin book):
If $left(X,tauright)$ is a topological space and $inftynotin X$, then $left(X_infty,tau_inftyright)$, where $X_infty=Xcupleftinftyright$ and every $Uintau_infty$ is such that $Uintau$ or $U^csubseteq X$ is compact, is a topological space.
I believe that this definition requires that $U^csubseteq X$ be compact and closed.
Edit: The first question was my attempt to show that if $U,Vintau_infty$ are such that $Uintau$ and $V^csubseteq X$ is compact, then $left(Ucup Vright)^c=U^ccap V^c$ is not compact. However, this holds, as the comments show. A correct counter-example would be to show that if $U,Vintau_infty$ are such that $U^c,V^csubseteq X$ are compact, then $left(Ucup Vright)^c=U^ccap V^c$ is not compact, as Rob Arthan shows in his answer.
general-topology
asked Aug 9 at 21:39
Cleric
3,06632463
7
$C cap K$ is a (relatively) closed subset of $K$, hence compact.
– Daniel Fischer♦
Aug 9 at 21:42
2
For any open cover $cal U$ of $C$, adding $Ksetminus C$ to $cal U$ gives an open cover of $K$. It must therefore have a finite subcover; removing $Ksetminus C$ from it gives a cover of $C$.
– anomaly
Aug 9 at 21:46
And yes, if you take a non-closed (quasi)compact $K subset X$, let $U = X_inftysetminus K$, then $U cap X notin tau_infty$. Does Rudin only consider Hausdorff spaces there?
– Daniel Fischer♦
Aug 9 at 21:46
3
Possible duplicate of Is the intersection of a closed set and a compact set always compact?
– zzuussee
Aug 9 at 21:48
1
@zzuussee: this is about topological spaces not metric spaces and so not a duplicate of the question you cite.
– Rob Arthan
Aug 9 at 21:49
 |Â
show 2 more comments
up vote
3
down vote
favorite
up vote
3
down vote
favorite
What is an example of a topological space $X$ such that $C,Ksubseteq X$; $C$ is closed; $K$ is compact; and $Ccap K$ is not compact?
I know that $X$ can be neither Hausdorff nor finite.
I am interested in this question because I recently read the following definition (in a Rudin book):
If $left(X,tauright)$ is a topological space and $inftynotin X$, then $left(X_infty,tau_inftyright)$, where $X_infty=Xcupleftinftyright$ and every $Uintau_infty$ is such that $Uintau$ or $U^csubseteq X$ is compact, is a topological space.
I believe that this definition requires that $U^csubseteq X$ be compact and closed.
Edit: The first question was my attempt to show that if $U,Vintau_infty$ are such that $Uintau$ and $V^csubseteq X$ is compact, then $left(Ucup Vright)^c=U^ccap V^c$ is not compact. However, this holds, as the comments show. A correct counter-example would be to show that if $U,Vintau_infty$ are such that $U^c,V^csubseteq X$ are compact, then $left(Ucup Vright)^c=U^ccap V^c$ is not compact, as Rob Arthan shows in his answer.
general-topology
asked Aug 9 at 21:39
Cleric
3,06632463
What is an example of a topological space $X$ such that $C,Ksubseteq X$; $C$ is closed; $K$ is compact; and $Ccap K$ is not compact?
I know that $X$ can be neither Hausdorff nor finite.
I am interested in this question because I recently read the following definition (in a Rudin book):
If $left(X,tauright)$ is a topological space and $inftynotin X$, then $left(X_infty,tau_inftyright)$, where $X_infty=Xcupleftinftyright$ and every $Uintau_infty$ is such that $Uintau$ or $U^csubseteq X$ is compact, is a topological space.
I believe that this definition requires that $U^csubseteq X$ be compact and closed.
Edit: The first question was my attempt to show that if $U,Vintau_infty$ are such that $Uintau$ and $V^csubseteq X$ is compact, then $left(Ucup Vright)^c=U^ccap V^c$ is not compact. However, this holds, as the comments show. A correct counter-example would be to show that if $U,Vintau_infty$ are such that $U^c,V^csubseteq X$ are compact, then $left(Ucup Vright)^c=U^ccap V^c$ is not compact, as Rob Arthan shows in his answer.
general-topology
asked Aug 9 at 21:39
Cleric
3,06632463
edited Aug 9 at 23:06
asked Aug 9 at 21:39
Cleric
3,06632463
asked Aug 9 at 21:39
Cleric
3,06632463
asked Aug 9 at 21:39
Cleric
3,06632463
3,06632463
7
$C cap K$ is a (relatively) closed subset of $K$, hence compact.
– Daniel Fischer♦
Aug 9 at 21:42
2
For any open cover $cal U$ of $C$, adding $Ksetminus C$ to $cal U$ gives an open cover of $K$. It must therefore have a finite subcover; removing $Ksetminus C$ from it gives a cover of $C$.
– anomaly
Aug 9 at 21:46
And yes, if you take a non-closed (quasi)compact $K subset X$, let $U = X_inftysetminus K$, then $U cap X notin tau_infty$. Does Rudin only consider Hausdorff spaces there?
– Daniel Fischer♦
Aug 9 at 21:46
3
Possible duplicate of Is the intersection of a closed set and a compact set always compact?
– zzuussee
Aug 9 at 21:48
1
@zzuussee: this is about topological spaces not metric spaces and so not a duplicate of the question you cite.
– Rob Arthan
Aug 9 at 21:49
 |Â
show 2 more comments
7
$C cap K$ is a (relatively) closed subset of $K$, hence compact.
– Daniel Fischer♦
Aug 9 at 21:42
2
For any open cover $cal U$ of $C$, adding $Ksetminus C$ to $cal U$ gives an open cover of $K$. It must therefore have a finite subcover; removing $Ksetminus C$ from it gives a cover of $C$.
– anomaly
Aug 9 at 21:46
And yes, if you take a non-closed (quasi)compact $K subset X$, let $U = X_inftysetminus K$, then $U cap X notin tau_infty$. Does Rudin only consider Hausdorff spaces there?
– Daniel Fischer♦
Aug 9 at 21:46
3
Possible duplicate of Is the intersection of a closed set and a compact set always compact?
– zzuussee
Aug 9 at 21:48
1
@zzuussee: this is about topological spaces not metric spaces and so not a duplicate of the question you cite.
– Rob Arthan
Aug 9 at 21:49
7
7
$C cap K$ is a (relatively) closed subset of $K$, hence compact.
– Daniel Fischer♦
Aug 9 at 21:42
$C cap K$ is a (relatively) closed subset of $K$, hence compact.
– Daniel Fischer♦
Aug 9 at 21:42
2
2
For any open cover $cal U$ of $C$, adding $Ksetminus C$ to $cal U$ gives an open cover of $K$. It must therefore have a finite subcover; removing $Ksetminus C$ from it gives a cover of $C$.
– anomaly
Aug 9 at 21:46
For any open cover $cal U$ of $C$, adding $Ksetminus C$ to $cal U$ gives an open cover of $K$. It must therefore have a finite subcover; removing $Ksetminus C$ from it gives a cover of $C$.
– anomaly
Aug 9 at 21:46
And yes, if you take a non-closed (quasi)compact $K subset X$, let $U = X_inftysetminus K$, then $U cap X notin tau_infty$. Does Rudin only consider Hausdorff spaces there?
– Daniel Fischer♦
Aug 9 at 21:46
And yes, if you take a non-closed (quasi)compact $K subset X$, let $U = X_inftysetminus K$, then $U cap X notin tau_infty$. Does Rudin only consider Hausdorff spaces there?
– Daniel Fischer♦
Aug 9 at 21:46
3
3
Possible duplicate of Is the intersection of a closed set and a compact set always compact?
– zzuussee
Aug 9 at 21:48
Possible duplicate of Is the intersection of a closed set and a compact set always compact?
– zzuussee
Aug 9 at 21:48
1
1
@zzuussee: this is about topological spaces not metric spaces and so not a duplicate of the question you cite.
– Rob Arthan
Aug 9 at 21:49
@zzuussee: this is about topological spaces not metric spaces and so not a duplicate of the question you cite.
– Rob Arthan
Aug 9 at 21:49
 |Â
show 2 more comments
1 Answer
1
active
oldest
votes
up vote
3
down vote
accepted
I'm reading this question as asking about the OP's belief in the last sentence as well as just about the sentence with the question mark at the beginning (which has been addressed in the comments: the intersection $C cap K$ of a closed set $C$ and a compact set $K$ is always compact).
In a space that isn't Hausdorff, compact sets aren't necessarily closed under intersections. E.g., take $(X, tau)$ to be the line with two origins: then (using a notation that I hope is obvious), $A = [0_a, 1]$ and $B = [0_b, 1]$ are both compact but $A cap B = (0_a, 1] = (0_b, 1]$ is not compact. So in Rudin's definition, you do indeed need to require $U^c$ to be compact and closed for the proposed set of open sets $tau_infty$ to satisfy the axioms of a topological space.
N.b., with the new MSE theme you can barely see it, but the words "line with two origins" above are a link to the Wikipedia page on that subject.
answered Aug 9 at 22:41
Rob Arthan
27.1k42864
Perhaps, now that I finally did bin my answer, it's worth just saying that there is indeed no example of what OP asks for (as in the comments to their question).
– Matt
Aug 9 at 22:52
@Matt: thanks for the reminder. I've edited my answer to address that.
– Rob Arthan
Aug 9 at 22:55
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
I'm reading this question as asking about the OP's belief in the last sentence as well as just about the sentence with the question mark at the beginning (which has been addressed in the comments: the intersection $C cap K$ of a closed set $C$ and a compact set $K$ is always compact).
In a space that isn't Hausdorff, compact sets aren't necessarily closed under intersections. E.g., take $(X, tau)$ to be the line with two origins: then (using a notation that I hope is obvious), $A = [0_a, 1]$ and $B = [0_b, 1]$ are both compact but $A cap B = (0_a, 1] = (0_b, 1]$ is not compact. So in Rudin's definition, you do indeed need to require $U^c$ to be compact and closed for the proposed set of open sets $tau_infty$ to satisfy the axioms of a topological space.
N.b., with the new MSE theme you can barely see it, but the words "line with two origins" above are a link to the Wikipedia page on that subject.
answered Aug 9 at 22:41
Rob Arthan
27.1k42864
Perhaps, now that I finally did bin my answer, it's worth just saying that there is indeed no example of what OP asks for (as in the comments to their question).
– Matt
Aug 9 at 22:52
@Matt: thanks for the reminder. I've edited my answer to address that.
– Rob Arthan
Aug 9 at 22:55
add a comment |Â
up vote
3
down vote
accepted
I'm reading this question as asking about the OP's belief in the last sentence as well as just about the sentence with the question mark at the beginning (which has been addressed in the comments: the intersection $C cap K$ of a closed set $C$ and a compact set $K$ is always compact).
In a space that isn't Hausdorff, compact sets aren't necessarily closed under intersections. E.g., take $(X, tau)$ to be the line with two origins: then (using a notation that I hope is obvious), $A = [0_a, 1]$ and $B = [0_b, 1]$ are both compact but $A cap B = (0_a, 1] = (0_b, 1]$ is not compact. So in Rudin's definition, you do indeed need to require $U^c$ to be compact and closed for the proposed set of open sets $tau_infty$ to satisfy the axioms of a topological space.
N.b., with the new MSE theme you can barely see it, but the words "line with two origins" above are a link to the Wikipedia page on that subject.
answered Aug 9 at 22:41
Rob Arthan
27.1k42864
Perhaps, now that I finally did bin my answer, it's worth just saying that there is indeed no example of what OP asks for (as in the comments to their question).
– Matt
Aug 9 at 22:52
@Matt: thanks for the reminder. I've edited my answer to address that.
– Rob Arthan
Aug 9 at 22:55
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
I'm reading this question as asking about the OP's belief in the last sentence as well as just about the sentence with the question mark at the beginning (which has been addressed in the comments: the intersection $C cap K$ of a closed set $C$ and a compact set $K$ is always compact).
In a space that isn't Hausdorff, compact sets aren't necessarily closed under intersections. E.g., take $(X, tau)$ to be the line with two origins: then (using a notation that I hope is obvious), $A = [0_a, 1]$ and $B = [0_b, 1]$ are both compact but $A cap B = (0_a, 1] = (0_b, 1]$ is not compact. So in Rudin's definition, you do indeed need to require $U^c$ to be compact and closed for the proposed set of open sets $tau_infty$ to satisfy the axioms of a topological space.
N.b., with the new MSE theme you can barely see it, but the words "line with two origins" above are a link to the Wikipedia page on that subject.
answered Aug 9 at 22:41
Rob Arthan
27.1k42864
I'm reading this question as asking about the OP's belief in the last sentence as well as just about the sentence with the question mark at the beginning (which has been addressed in the comments: the intersection $C cap K$ of a closed set $C$ and a compact set $K$ is always compact).
In a space that isn't Hausdorff, compact sets aren't necessarily closed under intersections. E.g., take $(X, tau)$ to be the line with two origins: then (using a notation that I hope is obvious), $A = [0_a, 1]$ and $B = [0_b, 1]$ are both compact but $A cap B = (0_a, 1] = (0_b, 1]$ is not compact. So in Rudin's definition, you do indeed need to require $U^c$ to be compact and closed for the proposed set of open sets $tau_infty$ to satisfy the axioms of a topological space.
N.b., with the new MSE theme you can barely see it, but the words "line with two origins" above are a link to the Wikipedia page on that subject.
answered Aug 9 at 22:41
Rob Arthan
27.1k42864
edited Aug 9 at 22:54
answered Aug 9 at 22:41
Rob Arthan
27.1k42864
answered Aug 9 at 22:41
Rob Arthan
27.1k42864
answered Aug 9 at 22:41
Rob Arthan
27.1k42864
27.1k42864
Perhaps, now that I finally did bin my answer, it's worth just saying that there is indeed no example of what OP asks for (as in the comments to their question).
– Matt
Aug 9 at 22:52
@Matt: thanks for the reminder. I've edited my answer to address that.
– Rob Arthan
Aug 9 at 22:55
add a comment |Â
Perhaps, now that I finally did bin my answer, it's worth just saying that there is indeed no example of what OP asks for (as in the comments to their question).
– Matt
Aug 9 at 22:52
@Matt: thanks for the reminder. I've edited my answer to address that.
– Rob Arthan
Aug 9 at 22:55
Perhaps, now that I finally did bin my answer, it's worth just saying that there is indeed no example of what OP asks for (as in the comments to their question).
– Matt
Aug 9 at 22:52
Perhaps, now that I finally did bin my answer, it's worth just saying that there is indeed no example of what OP asks for (as in the comments to their question).
– Matt
Aug 9 at 22:52
@Matt: thanks for the reminder. I've edited my answer to address that.
– Rob Arthan
Aug 9 at 22:55
@Matt: thanks for the reminder. I've edited my answer to address that.
– Rob Arthan
Aug 9 at 22:55
add a comment |Â
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7
$C cap K$ is a (relatively) closed subset of $K$, hence compact.
– Daniel Fischer♦
Aug 9 at 21:42
2
For any open cover $cal U$ of $C$, adding $Ksetminus C$ to $cal U$ gives an open cover of $K$. It must therefore have a finite subcover; removing $Ksetminus C$ from it gives a cover of $C$.
– anomaly
Aug 9 at 21:46
And yes, if you take a non-closed (quasi)compact $K subset X$, let $U = X_inftysetminus K$, then $U cap X notin tau_infty$. Does Rudin only consider Hausdorff spaces there?
– Daniel Fischer♦
Aug 9 at 21:46
3
Possible duplicate of Is the intersection of a closed set and a compact set always compact?
– zzuussee
Aug 9 at 21:48
1
@zzuussee: this is about topological spaces not metric spaces and so not a duplicate of the question you cite.
– Rob Arthan
Aug 9 at 21:49