Continuous function on Hausdorff space
Clash Royale CLAN TAG#URR8PPP
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I'm wondering about how to solve this one:
Let $X$ be a topological Hausdorff space and $A$ a subspace of $X$.
Let $f:X rightarrow A$ continuous such that $f(a) = a hspace5mm forall a in A$.
Show that $A$ is closed.
Any ideas? I think I might get the solution just by adding compactness to hypoteses..
general-topology
asked 3 hours ago
James Arten
538
add a comment |Â
up vote
1
down vote
favorite
I'm wondering about how to solve this one:
Let $X$ be a topological Hausdorff space and $A$ a subspace of $X$.
Let $f:X rightarrow A$ continuous such that $f(a) = a hspace5mm forall a in A$.
Show that $A$ is closed.
Any ideas? I think I might get the solution just by adding compactness to hypoteses..
general-topology
asked 3 hours ago
James Arten
538
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm wondering about how to solve this one:
Let $X$ be a topological Hausdorff space and $A$ a subspace of $X$.
Let $f:X rightarrow A$ continuous such that $f(a) = a hspace5mm forall a in A$.
Show that $A$ is closed.
Any ideas? I think I might get the solution just by adding compactness to hypoteses..
general-topology
asked 3 hours ago
James Arten
538
I'm wondering about how to solve this one:
Let $X$ be a topological Hausdorff space and $A$ a subspace of $X$.
Let $f:X rightarrow A$ continuous such that $f(a) = a hspace5mm forall a in A$.
Show that $A$ is closed.
Any ideas? I think I might get the solution just by adding compactness to hypoteses..
general-topology
general-topology
asked 3 hours ago
James Arten
538
asked 3 hours ago
James Arten
538
asked 3 hours ago
James Arten
538
asked 3 hours ago
James Arten
538
asked 3 hours ago
James Arten
538
538
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
5
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Hint: Consider the map $g := (f, operatornameid) : X to X times X$. If $Delta_X$ is the diagonal in $X times X$ (which is closed by the hypothesis that $X$ is Hausdorff) then what can you say about $g^-1(Delta_X)$?
answered 3 hours ago
Daniel Schepler
7,7111617
$g^-1(Delta _X)$ is closed because g is continuous.. but could you explain me a little better? What do you mean by $(f,id)$?
– James Arten
3 hours ago
Also why diagonal is closed? And how it is defined?
– James Arten
3 hours ago
Another way of saying the definition of $g$ would be: $g(x) = (f(x), x)$.
– Daniel Schepler
3 hours ago
@JamesArten Have a look at this
– Dog_69
3 hours ago
The diagonal is $ (x,y) in X times X mid x = y $. If $X$ is Hausdorff, and $(x, y) in (Delta_X)^c$, that means that $x ne y$, so then choosing disjoint open neighborhoods $U,V$ of $x,y$, then we will have $(x, y) in U times V$ and $U times V subseteq (Delta_X)^c$.
– Daniel Schepler
3 hours ago
 |Â
show 1 more comment
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
Hint: Consider the map $g := (f, operatornameid) : X to X times X$. If $Delta_X$ is the diagonal in $X times X$ (which is closed by the hypothesis that $X$ is Hausdorff) then what can you say about $g^-1(Delta_X)$?
answered 3 hours ago
Daniel Schepler
7,7111617
$g^-1(Delta _X)$ is closed because g is continuous.. but could you explain me a little better? What do you mean by $(f,id)$?
– James Arten
3 hours ago
Also why diagonal is closed? And how it is defined?
– James Arten
3 hours ago
Another way of saying the definition of $g$ would be: $g(x) = (f(x), x)$.
– Daniel Schepler
3 hours ago
@JamesArten Have a look at this
– Dog_69
3 hours ago
The diagonal is $ (x,y) in X times X mid x = y $. If $X$ is Hausdorff, and $(x, y) in (Delta_X)^c$, that means that $x ne y$, so then choosing disjoint open neighborhoods $U,V$ of $x,y$, then we will have $(x, y) in U times V$ and $U times V subseteq (Delta_X)^c$.
– Daniel Schepler
3 hours ago
 |Â
show 1 more comment
up vote
5
down vote
Hint: Consider the map $g := (f, operatornameid) : X to X times X$. If $Delta_X$ is the diagonal in $X times X$ (which is closed by the hypothesis that $X$ is Hausdorff) then what can you say about $g^-1(Delta_X)$?
answered 3 hours ago
Daniel Schepler
7,7111617
$g^-1(Delta _X)$ is closed because g is continuous.. but could you explain me a little better? What do you mean by $(f,id)$?
– James Arten
3 hours ago
Also why diagonal is closed? And how it is defined?
– James Arten
3 hours ago
Another way of saying the definition of $g$ would be: $g(x) = (f(x), x)$.
– Daniel Schepler
3 hours ago
@JamesArten Have a look at this
– Dog_69
3 hours ago
The diagonal is $ (x,y) in X times X mid x = y $. If $X$ is Hausdorff, and $(x, y) in (Delta_X)^c$, that means that $x ne y$, so then choosing disjoint open neighborhoods $U,V$ of $x,y$, then we will have $(x, y) in U times V$ and $U times V subseteq (Delta_X)^c$.
– Daniel Schepler
3 hours ago
 |Â
show 1 more comment
up vote
5
down vote
up vote
5
down vote
Hint: Consider the map $g := (f, operatornameid) : X to X times X$. If $Delta_X$ is the diagonal in $X times X$ (which is closed by the hypothesis that $X$ is Hausdorff) then what can you say about $g^-1(Delta_X)$?
answered 3 hours ago
Daniel Schepler
7,7111617
Hint: Consider the map $g := (f, operatornameid) : X to X times X$. If $Delta_X$ is the diagonal in $X times X$ (which is closed by the hypothesis that $X$ is Hausdorff) then what can you say about $g^-1(Delta_X)$?
answered 3 hours ago
Daniel Schepler
7,7111617
answered 3 hours ago
Daniel Schepler
7,7111617
answered 3 hours ago
Daniel Schepler
7,7111617
answered 3 hours ago
Daniel Schepler
7,7111617
7,7111617
$g^-1(Delta _X)$ is closed because g is continuous.. but could you explain me a little better? What do you mean by $(f,id)$?
– James Arten
3 hours ago
Also why diagonal is closed? And how it is defined?
– James Arten
3 hours ago
Another way of saying the definition of $g$ would be: $g(x) = (f(x), x)$.
– Daniel Schepler
3 hours ago
@JamesArten Have a look at this
– Dog_69
3 hours ago
The diagonal is $ (x,y) in X times X mid x = y $. If $X$ is Hausdorff, and $(x, y) in (Delta_X)^c$, that means that $x ne y$, so then choosing disjoint open neighborhoods $U,V$ of $x,y$, then we will have $(x, y) in U times V$ and $U times V subseteq (Delta_X)^c$.
– Daniel Schepler
3 hours ago
 |Â
show 1 more comment
$g^-1(Delta _X)$ is closed because g is continuous.. but could you explain me a little better? What do you mean by $(f,id)$?
– James Arten
3 hours ago
Also why diagonal is closed? And how it is defined?
– James Arten
3 hours ago
Another way of saying the definition of $g$ would be: $g(x) = (f(x), x)$.
– Daniel Schepler
3 hours ago
@JamesArten Have a look at this
– Dog_69
3 hours ago
The diagonal is $ (x,y) in X times X mid x = y $. If $X$ is Hausdorff, and $(x, y) in (Delta_X)^c$, that means that $x ne y$, so then choosing disjoint open neighborhoods $U,V$ of $x,y$, then we will have $(x, y) in U times V$ and $U times V subseteq (Delta_X)^c$.
– Daniel Schepler
3 hours ago
$g^-1(Delta _X)$ is closed because g is continuous.. but could you explain me a little better? What do you mean by $(f,id)$?
– James Arten
3 hours ago
$g^-1(Delta _X)$ is closed because g is continuous.. but could you explain me a little better? What do you mean by $(f,id)$?
– James Arten
3 hours ago
Also why diagonal is closed? And how it is defined?
– James Arten
3 hours ago
Also why diagonal is closed? And how it is defined?
– James Arten
3 hours ago
Another way of saying the definition of $g$ would be: $g(x) = (f(x), x)$.
– Daniel Schepler
3 hours ago
Another way of saying the definition of $g$ would be: $g(x) = (f(x), x)$.
– Daniel Schepler
3 hours ago
@JamesArten Have a look at this
– Dog_69
3 hours ago
@JamesArten Have a look at this
– Dog_69
3 hours ago
The diagonal is $ (x,y) in X times X mid x = y $. If $X$ is Hausdorff, and $(x, y) in (Delta_X)^c$, that means that $x ne y$, so then choosing disjoint open neighborhoods $U,V$ of $x,y$, then we will have $(x, y) in U times V$ and $U times V subseteq (Delta_X)^c$.
– Daniel Schepler
3 hours ago
The diagonal is $ (x,y) in X times X mid x = y $. If $X$ is Hausdorff, and $(x, y) in (Delta_X)^c$, that means that $x ne y$, so then choosing disjoint open neighborhoods $U,V$ of $x,y$, then we will have $(x, y) in U times V$ and $U times V subseteq (Delta_X)^c$.
– Daniel Schepler
3 hours ago
 |Â
show 1 more comment
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