Equivalence of definitions of a closed set
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So I am working through some problems on my own and ran into one I have a question about. It is as follows. Let $(X_1,tau_1)$ and $(X_2,tau_2)$ be topological spaces, show that a set is closed iff whenever ${x_gamma}_{gammainGamma}subseteq A$ with $x_gammarightarrow xin X$ then $xin A.$ I don't see how to use the hint that $mathcal{U}(x)={Uintau_1:xin U}$ form a directed set with $Uleq V$ implying $Vsubseteq U$. Any help pointing me in the right direction would be appreciated.
Attempt: Suppose that whenever ${x_gamma}_{gammainGamma}subseteq A$ with $x_gammarightarrow xin X$ then $xin A.$ For all $xin X_1$ s.t. $xnotin A$
(i.e. $xin A^C$) we have that $x$ is not a limit point of $A$. Hence, $A^C$ is a neighborhood of $x$. Now we only need that $A^C$ is open, which is true since it is a neighborhood of all its points as $x$ was arbitrary. This should give us the first direction. I don't see how to get the other direction based using the provided hint.
general-topology
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|
show 1 more comment
$begingroup$
So I am working through some problems on my own and ran into one I have a question about. It is as follows. Let $(X_1,tau_1)$ and $(X_2,tau_2)$ be topological spaces, show that a set is closed iff whenever ${x_gamma}_{gammainGamma}subseteq A$ with $x_gammarightarrow xin X$ then $xin A.$ I don't see how to use the hint that $mathcal{U}(x)={Uintau_1:xin U}$ form a directed set with $Uleq V$ implying $Vsubseteq U$. Any help pointing me in the right direction would be appreciated.
Attempt: Suppose that whenever ${x_gamma}_{gammainGamma}subseteq A$ with $x_gammarightarrow xin X$ then $xin A.$ For all $xin X_1$ s.t. $xnotin A$
(i.e. $xin A^C$) we have that $x$ is not a limit point of $A$. Hence, $A^C$ is a neighborhood of $x$. Now we only need that $A^C$ is open, which is true since it is a neighborhood of all its points as $x$ was arbitrary. This should give us the first direction. I don't see how to get the other direction based using the provided hint.
general-topology
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$begingroup$
What have you tried so far? What is your definition of closed?
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– Dave
Dec 31 '18 at 1:09
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I'm having trouble seeing where to start. I am using the definition that the complement is open.
$endgroup$
– Scott
Dec 31 '18 at 1:13
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Is your definition of closed the following: A set is closed iff it’s complement is open. If this is the definition you’re using then it’ll be quite easy to show what you want.
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– John Mitchell
Dec 31 '18 at 1:13
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I know it shouldn't be difficult, but for some reason it just isn't clicking...
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– Scott
Dec 31 '18 at 1:22
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It seems that $X_1$ and $X_2$ play no role in the problem.
$endgroup$
– Keenan Kidwell
Dec 31 '18 at 2:25
|
show 1 more comment
$begingroup$
So I am working through some problems on my own and ran into one I have a question about. It is as follows. Let $(X_1,tau_1)$ and $(X_2,tau_2)$ be topological spaces, show that a set is closed iff whenever ${x_gamma}_{gammainGamma}subseteq A$ with $x_gammarightarrow xin X$ then $xin A.$ I don't see how to use the hint that $mathcal{U}(x)={Uintau_1:xin U}$ form a directed set with $Uleq V$ implying $Vsubseteq U$. Any help pointing me in the right direction would be appreciated.
Attempt: Suppose that whenever ${x_gamma}_{gammainGamma}subseteq A$ with $x_gammarightarrow xin X$ then $xin A.$ For all $xin X_1$ s.t. $xnotin A$
(i.e. $xin A^C$) we have that $x$ is not a limit point of $A$. Hence, $A^C$ is a neighborhood of $x$. Now we only need that $A^C$ is open, which is true since it is a neighborhood of all its points as $x$ was arbitrary. This should give us the first direction. I don't see how to get the other direction based using the provided hint.
general-topology
$endgroup$
So I am working through some problems on my own and ran into one I have a question about. It is as follows. Let $(X_1,tau_1)$ and $(X_2,tau_2)$ be topological spaces, show that a set is closed iff whenever ${x_gamma}_{gammainGamma}subseteq A$ with $x_gammarightarrow xin X$ then $xin A.$ I don't see how to use the hint that $mathcal{U}(x)={Uintau_1:xin U}$ form a directed set with $Uleq V$ implying $Vsubseteq U$. Any help pointing me in the right direction would be appreciated.
Attempt: Suppose that whenever ${x_gamma}_{gammainGamma}subseteq A$ with $x_gammarightarrow xin X$ then $xin A.$ For all $xin X_1$ s.t. $xnotin A$
(i.e. $xin A^C$) we have that $x$ is not a limit point of $A$. Hence, $A^C$ is a neighborhood of $x$. Now we only need that $A^C$ is open, which is true since it is a neighborhood of all its points as $x$ was arbitrary. This should give us the first direction. I don't see how to get the other direction based using the provided hint.
general-topology
general-topology
edited Dec 31 '18 at 1:34
Scott
asked Dec 31 '18 at 1:08
ScottScott
38918
38918
$begingroup$
What have you tried so far? What is your definition of closed?
$endgroup$
– Dave
Dec 31 '18 at 1:09
$begingroup$
I'm having trouble seeing where to start. I am using the definition that the complement is open.
$endgroup$
– Scott
Dec 31 '18 at 1:13
$begingroup$
Is your definition of closed the following: A set is closed iff it’s complement is open. If this is the definition you’re using then it’ll be quite easy to show what you want.
$endgroup$
– John Mitchell
Dec 31 '18 at 1:13
$begingroup$
I know it shouldn't be difficult, but for some reason it just isn't clicking...
$endgroup$
– Scott
Dec 31 '18 at 1:22
$begingroup$
It seems that $X_1$ and $X_2$ play no role in the problem.
$endgroup$
– Keenan Kidwell
Dec 31 '18 at 2:25
|
show 1 more comment
$begingroup$
What have you tried so far? What is your definition of closed?
$endgroup$
– Dave
Dec 31 '18 at 1:09
$begingroup$
I'm having trouble seeing where to start. I am using the definition that the complement is open.
$endgroup$
– Scott
Dec 31 '18 at 1:13
$begingroup$
Is your definition of closed the following: A set is closed iff it’s complement is open. If this is the definition you’re using then it’ll be quite easy to show what you want.
$endgroup$
– John Mitchell
Dec 31 '18 at 1:13
$begingroup$
I know it shouldn't be difficult, but for some reason it just isn't clicking...
$endgroup$
– Scott
Dec 31 '18 at 1:22
$begingroup$
It seems that $X_1$ and $X_2$ play no role in the problem.
$endgroup$
– Keenan Kidwell
Dec 31 '18 at 2:25
$begingroup$
What have you tried so far? What is your definition of closed?
$endgroup$
– Dave
Dec 31 '18 at 1:09
$begingroup$
What have you tried so far? What is your definition of closed?
$endgroup$
– Dave
Dec 31 '18 at 1:09
$begingroup$
I'm having trouble seeing where to start. I am using the definition that the complement is open.
$endgroup$
– Scott
Dec 31 '18 at 1:13
$begingroup$
I'm having trouble seeing where to start. I am using the definition that the complement is open.
$endgroup$
– Scott
Dec 31 '18 at 1:13
$begingroup$
Is your definition of closed the following: A set is closed iff it’s complement is open. If this is the definition you’re using then it’ll be quite easy to show what you want.
$endgroup$
– John Mitchell
Dec 31 '18 at 1:13
$begingroup$
Is your definition of closed the following: A set is closed iff it’s complement is open. If this is the definition you’re using then it’ll be quite easy to show what you want.
$endgroup$
– John Mitchell
Dec 31 '18 at 1:13
$begingroup$
I know it shouldn't be difficult, but for some reason it just isn't clicking...
$endgroup$
– Scott
Dec 31 '18 at 1:22
$begingroup$
I know it shouldn't be difficult, but for some reason it just isn't clicking...
$endgroup$
– Scott
Dec 31 '18 at 1:22
$begingroup$
It seems that $X_1$ and $X_2$ play no role in the problem.
$endgroup$
– Keenan Kidwell
Dec 31 '18 at 2:25
$begingroup$
It seems that $X_1$ and $X_2$ play no role in the problem.
$endgroup$
– Keenan Kidwell
Dec 31 '18 at 2:25
|
show 1 more comment
2 Answers
2
active
oldest
votes
$begingroup$
Assume A closed. Let n be a net into A that converges to a.
To show a in A use the theorem for closed A
x in A iff for all open U nhood x, U $cap$ A not empty
and some facts about convergence of nets to show x in A.
Conversely, to prove A is closed, show
if for all open U nhood x, U $cap$ A not empty, then x in A.
So assume for all open U nhood x, U $cap$ A not empty.
Use the hint to construct a net into A that converges to x
and with that conclude x in A.
$endgroup$
add a comment |
$begingroup$
You define $C$ to be closed if it’s complement is open. But the net convergence equivalent is easier proved using the following equivalent definition:
$C$ is closed iff for all $x in X$: if every open set $O$ that contains $x$ intersects $C$, then $x in C$.
Given this, suppose $C$ obeys the net convergence condition. Suppose that $x$ is a point such that every open neighbourhood of $x$ intersects $C$. Then we can apply the hint: define $Gamma$ be the directed set $mathcal{U}(x)$ under reverse inclusion, and pick (by the assumption on $x$ this is possible) a point $x(U) in C cap U$ for every $U in mathcal{U}(x)$. Then this defined a net: $x: Gamma to X$ where all values lie in $C$ by construction and also $x_gamma to x$: let $O$ be any open subset that contains $x$ and note that $O in Gamma$ and if $U ge O$ we have that $U subseteq O$ and so $x(U) in U subseteq O$, so $x(U) in O$. So this shows convergence. By the net convergence condition we can thus conclude $x in C$ and we are done with one direction.
Suppose $C$ is closed and let $x: Gamma to X$ be any net with values in $C$ such that it converges to $p in X$. Then every open neighbourhood of $p$ contains a point of the net (by convergence to $p$) so in particular a point of $C$. By the equivalent formulation of closedness we see that $p in C$ and so $C$ obeys the net convergence condition.
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add a comment |
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2 Answers
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2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Assume A closed. Let n be a net into A that converges to a.
To show a in A use the theorem for closed A
x in A iff for all open U nhood x, U $cap$ A not empty
and some facts about convergence of nets to show x in A.
Conversely, to prove A is closed, show
if for all open U nhood x, U $cap$ A not empty, then x in A.
So assume for all open U nhood x, U $cap$ A not empty.
Use the hint to construct a net into A that converges to x
and with that conclude x in A.
$endgroup$
add a comment |
$begingroup$
Assume A closed. Let n be a net into A that converges to a.
To show a in A use the theorem for closed A
x in A iff for all open U nhood x, U $cap$ A not empty
and some facts about convergence of nets to show x in A.
Conversely, to prove A is closed, show
if for all open U nhood x, U $cap$ A not empty, then x in A.
So assume for all open U nhood x, U $cap$ A not empty.
Use the hint to construct a net into A that converges to x
and with that conclude x in A.
$endgroup$
add a comment |
$begingroup$
Assume A closed. Let n be a net into A that converges to a.
To show a in A use the theorem for closed A
x in A iff for all open U nhood x, U $cap$ A not empty
and some facts about convergence of nets to show x in A.
Conversely, to prove A is closed, show
if for all open U nhood x, U $cap$ A not empty, then x in A.
So assume for all open U nhood x, U $cap$ A not empty.
Use the hint to construct a net into A that converges to x
and with that conclude x in A.
$endgroup$
Assume A closed. Let n be a net into A that converges to a.
To show a in A use the theorem for closed A
x in A iff for all open U nhood x, U $cap$ A not empty
and some facts about convergence of nets to show x in A.
Conversely, to prove A is closed, show
if for all open U nhood x, U $cap$ A not empty, then x in A.
So assume for all open U nhood x, U $cap$ A not empty.
Use the hint to construct a net into A that converges to x
and with that conclude x in A.
answered Dec 31 '18 at 1:59
William ElliotWilliam Elliot
8,8232820
8,8232820
add a comment |
add a comment |
$begingroup$
You define $C$ to be closed if it’s complement is open. But the net convergence equivalent is easier proved using the following equivalent definition:
$C$ is closed iff for all $x in X$: if every open set $O$ that contains $x$ intersects $C$, then $x in C$.
Given this, suppose $C$ obeys the net convergence condition. Suppose that $x$ is a point such that every open neighbourhood of $x$ intersects $C$. Then we can apply the hint: define $Gamma$ be the directed set $mathcal{U}(x)$ under reverse inclusion, and pick (by the assumption on $x$ this is possible) a point $x(U) in C cap U$ for every $U in mathcal{U}(x)$. Then this defined a net: $x: Gamma to X$ where all values lie in $C$ by construction and also $x_gamma to x$: let $O$ be any open subset that contains $x$ and note that $O in Gamma$ and if $U ge O$ we have that $U subseteq O$ and so $x(U) in U subseteq O$, so $x(U) in O$. So this shows convergence. By the net convergence condition we can thus conclude $x in C$ and we are done with one direction.
Suppose $C$ is closed and let $x: Gamma to X$ be any net with values in $C$ such that it converges to $p in X$. Then every open neighbourhood of $p$ contains a point of the net (by convergence to $p$) so in particular a point of $C$. By the equivalent formulation of closedness we see that $p in C$ and so $C$ obeys the net convergence condition.
$endgroup$
add a comment |
$begingroup$
You define $C$ to be closed if it’s complement is open. But the net convergence equivalent is easier proved using the following equivalent definition:
$C$ is closed iff for all $x in X$: if every open set $O$ that contains $x$ intersects $C$, then $x in C$.
Given this, suppose $C$ obeys the net convergence condition. Suppose that $x$ is a point such that every open neighbourhood of $x$ intersects $C$. Then we can apply the hint: define $Gamma$ be the directed set $mathcal{U}(x)$ under reverse inclusion, and pick (by the assumption on $x$ this is possible) a point $x(U) in C cap U$ for every $U in mathcal{U}(x)$. Then this defined a net: $x: Gamma to X$ where all values lie in $C$ by construction and also $x_gamma to x$: let $O$ be any open subset that contains $x$ and note that $O in Gamma$ and if $U ge O$ we have that $U subseteq O$ and so $x(U) in U subseteq O$, so $x(U) in O$. So this shows convergence. By the net convergence condition we can thus conclude $x in C$ and we are done with one direction.
Suppose $C$ is closed and let $x: Gamma to X$ be any net with values in $C$ such that it converges to $p in X$. Then every open neighbourhood of $p$ contains a point of the net (by convergence to $p$) so in particular a point of $C$. By the equivalent formulation of closedness we see that $p in C$ and so $C$ obeys the net convergence condition.
$endgroup$
add a comment |
$begingroup$
You define $C$ to be closed if it’s complement is open. But the net convergence equivalent is easier proved using the following equivalent definition:
$C$ is closed iff for all $x in X$: if every open set $O$ that contains $x$ intersects $C$, then $x in C$.
Given this, suppose $C$ obeys the net convergence condition. Suppose that $x$ is a point such that every open neighbourhood of $x$ intersects $C$. Then we can apply the hint: define $Gamma$ be the directed set $mathcal{U}(x)$ under reverse inclusion, and pick (by the assumption on $x$ this is possible) a point $x(U) in C cap U$ for every $U in mathcal{U}(x)$. Then this defined a net: $x: Gamma to X$ where all values lie in $C$ by construction and also $x_gamma to x$: let $O$ be any open subset that contains $x$ and note that $O in Gamma$ and if $U ge O$ we have that $U subseteq O$ and so $x(U) in U subseteq O$, so $x(U) in O$. So this shows convergence. By the net convergence condition we can thus conclude $x in C$ and we are done with one direction.
Suppose $C$ is closed and let $x: Gamma to X$ be any net with values in $C$ such that it converges to $p in X$. Then every open neighbourhood of $p$ contains a point of the net (by convergence to $p$) so in particular a point of $C$. By the equivalent formulation of closedness we see that $p in C$ and so $C$ obeys the net convergence condition.
$endgroup$
You define $C$ to be closed if it’s complement is open. But the net convergence equivalent is easier proved using the following equivalent definition:
$C$ is closed iff for all $x in X$: if every open set $O$ that contains $x$ intersects $C$, then $x in C$.
Given this, suppose $C$ obeys the net convergence condition. Suppose that $x$ is a point such that every open neighbourhood of $x$ intersects $C$. Then we can apply the hint: define $Gamma$ be the directed set $mathcal{U}(x)$ under reverse inclusion, and pick (by the assumption on $x$ this is possible) a point $x(U) in C cap U$ for every $U in mathcal{U}(x)$. Then this defined a net: $x: Gamma to X$ where all values lie in $C$ by construction and also $x_gamma to x$: let $O$ be any open subset that contains $x$ and note that $O in Gamma$ and if $U ge O$ we have that $U subseteq O$ and so $x(U) in U subseteq O$, so $x(U) in O$. So this shows convergence. By the net convergence condition we can thus conclude $x in C$ and we are done with one direction.
Suppose $C$ is closed and let $x: Gamma to X$ be any net with values in $C$ such that it converges to $p in X$. Then every open neighbourhood of $p$ contains a point of the net (by convergence to $p$) so in particular a point of $C$. By the equivalent formulation of closedness we see that $p in C$ and so $C$ obeys the net convergence condition.
answered Dec 31 '18 at 13:07
Henno BrandsmaHenno Brandsma
114k348123
114k348123
add a comment |
add a comment |
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$begingroup$
What have you tried so far? What is your definition of closed?
$endgroup$
– Dave
Dec 31 '18 at 1:09
$begingroup$
I'm having trouble seeing where to start. I am using the definition that the complement is open.
$endgroup$
– Scott
Dec 31 '18 at 1:13
$begingroup$
Is your definition of closed the following: A set is closed iff it’s complement is open. If this is the definition you’re using then it’ll be quite easy to show what you want.
$endgroup$
– John Mitchell
Dec 31 '18 at 1:13
$begingroup$
I know it shouldn't be difficult, but for some reason it just isn't clicking...
$endgroup$
– Scott
Dec 31 '18 at 1:22
$begingroup$
It seems that $X_1$ and $X_2$ play no role in the problem.
$endgroup$
– Keenan Kidwell
Dec 31 '18 at 2:25