Topology question

**Defunkt** · March 1st 2010, 01:23 PM

Hi, I would appreciate some guiding with this question. Thanks.

Let X be some topological space and Y a compact, Hausdorff topological space. Let $f:X \to Y$ and define $\Gamma_f = \{(x, f(x)) : x \in X\}$ .

Show that if $\Gamma_f$ is a closed subset of $X \times Y$ with the product topology, then f is continuous.

**Drexel28** · March 1st 2010, 01:36 PM

Originally Posted by Defunkt

Hi, I would appreciate some guiding with this question. Thanks.

Let X be some topological space and Y a compact, Hausdorff topological space. Let $f:X \to Y$ and define $\Gamma_f = \{(x, f(x)) : x \in X\}$ .

Show that if $\Gamma_f$ is a closed subset of $X \times Y$ with the product topology, then f is continuous.

What have you tried?

P.S. $Y$ need not be Hausdorff.

**Drexel28** · March 1st 2010, 02:01 PM

Theorem: Let $X$ be any space and $Y$ be compact. Then, if $\Gamma_f\subseteq X\times Y$ is closed we have that $f:X\mapsto Y$ is continuous.

Proof:

Lemma: Let $Y$ be compact, then $\pi:X\times Y\mapsto X$ is a closed.
Proof: Let $E\subseteq X\times Y$ be closed. We prove that $\pi(E)$ is closed by proving that it's compliment is open. So, let $x\notin\pi(E)$ . Then, it follows that $\left(x\times Y\right)\cap E=\varnothing$ , and for each $y\in Y$ $(x,y)$ has a basic open neighborhood $G_x(y)\times F_y\subseteq X\times Y-E$ . Since the tube $x\times Y$ is compact (since it's homeomorphic to $Y$ ) there is a finite subcovering $\left\{U_{y_1},\cdots,U_{y_m}\right\}$ . The neighborhood $U=\bigcap_{k=1}^{m}U_x(y_k)$ is clearly disjoint from $\pi(E)$ . The conclusion follows. $\blacksquare$

Now, it is easily proved that since $\pi:X\times Y\mapsto X$ is closed so is $\pi\mid \Gamma_f:\Gamma_f\mapsto X$ . But, it is clearly a bijective continuous mapping and since it's closed it's also a homeomorphism. The problem follows after considering the following lemma

Lemma: A mapping $f:X\mapsto Y$ is continuous if $\pi\mid \Gamma_f:\Gamma_f\mapsto X$ is a homeomorphism.
Proof: We must merely note that $f=\pi_Y\circ\left(\pi_X\mid \Gamma_f\right)^{-1}$ .

**Defunkt** · March 1st 2010, 02:05 PM

Originally Posted by Drexel28

What have you tried?

P.S. $Y$ need not be Hausdorff.

I think I came up with a solution. Tell me if you see anything wrong please.

I want to show that f is continuous, ie. that if $V \subset Y$ is open then $f^{-1}(V)$ is open as well.

Let V be an open subset of Y. Let $U = V^c$ , then U is closed, and so is $X \times U$ . Since $\Gamma_f$ is closed as well, $\Gamma_f \cap (X \times U)$ is also closed.

From a previous result, we know that the projection $\pi_x:X \times Y \to X$ is a closed mapping when Y is compact. Therefore, $\pi_x(\Gamma_f \cap (X \times U) )$ is closed.

However, note that $\pi_x(\Gamma_f \cap (X \times U) )$ is exactly the set of points in X which f does not map into V, and it is closed, therefore $f^{-1}(V)$ is open and we are done.

I know it is a bit informal, but that is simply the outline.

Thanks again for helping.

**Drexel28** · March 1st 2010, 02:16 PM

Originally Posted by Defunkt

I think I came up with a solution. Tell me if you see anything wrong please.

I want to show that f is continuous, ie. that if $V \subset Y$ is open then $f^{-1}(V)$ is open as well.

Let V be an open subset of Y. Let $U = V^c$ , then U is closed, and so is $X \times U$ . Since $G$ is closed as well, $G \cap (X \times U)$ is also closed.

From a previous result, we know that the projection $\pi_x:X \times Y \to X$ is a closed mapping when Y is compact. Therefore, $\pi_x(G \cap (X \times U) )$ is closed.

However, note that $\pi_x(G \cap (X \times U) )$ is exactly the set of points in X which f does not map into V, and it is closed, therefore $f^{-1}(V)$ is open and we are done.

I know it is a bit informal, but that is simply the outline.

Thanks again for helping.

I'm having a little trouble following it, but I think you are saying that $\left[f^{-1}(V)\right]'=\pi_X\left(\Gamma_f\cap\left(X\times V'\right)\right)$ . If you can prove that I'll believe you.

**Defunkt** · March 1st 2010, 02:29 PM

Originally Posted by Drexel28

I'm having a little trouble following it, but I think you are saying that $\left[f^{-1}(V)\right]'=\pi_X\left(\Gamma_f\cap\left(X\times V'\right)\right)$ . If you can prove that I'll believe you.

$\Gamma_f \cap (X \times V') = \{(x, f(x)) : x \in X\} \cap \{(x,y) : y \notin V\} = \{(x, f(x)) : f(x) \notin V\} =$ $\{(x, f(x)) : x \notin f^{-1}(V)\}$

Sorry for being messy, by the way I mistakenly wrote G in my first answer instead of $\Gamma_f$ , but I fixed it now.

**Drexel28** · March 1st 2010, 02:31 PM

Originally Posted by Defunkt

$\Gamma_f \cap (X \times V') = \{(x, f(x)) : x \in X\} \cap \{(x,y) : y \notin V\} = \{(x, f(x)) : f(x) \notin V\} =$ $\{(x, f(x)) : x \notin f^{-1}(V)\}$

Sorry for being messy, by the way I mistakenly wrote G in my first answer instead of $\Gamma_f$ , but I fixed it now.

Then I would agree!

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