Connected Hausdorff spaces and graphs

**Drexel28** · March 11th 2010, 04:29 PM

I came up with the following question, and would appreciate if I could get some validation/invalidation.

If $X$ is connected Hausdorff and $\varphi:X\mapsto Y$ continuous then is $\Gamma_\varphi\subseteq X\times Y$ connected? ( $\Gamma_\varphi$ is the graph of $\varphi$ ). I think the answer is yes.

Proof: We first need a lemma

Lemma: Let $X$ be connected Hausdorff, then $\Delta_X\subseteq X\times X$ is connected ( $\Delta_X$ is the diagonal).
Proof: Clearly the map $\iota_X\oplus\iota_X:X\mapsto X\times X$ given by $x\mapsto (x,x)$ is continuous. But, it is not hard to see that $\Delta_X=\left(\iota_X\oplus \iota_X\right)\left(X\right)$ . But, connectedness is invariant under continuous mapping. So, the conclusion follows $\blacksquare$

Now, it is easy to prove that the product of two continuous maps is continuous. In particular, since $\varphi$ and the identity map $\iota_X$ are continuous we have that $\iota_X\times\varphi:X\times X\mapsto X\times Y$ is continuous. I now claim that $\Gamma_\varphi=\left(\iota_X\times\varphi\right)\l eft(\Delta_X\right)$ .

To see this, let $(y,y')\in\Gamma_\varphi$ then $(y,y')=(x,\varphi(x))$ for some $x\in X$ . Clearly then $\left(\iota_X\times\varphi\right)\left((x,x)\right )=\left(x,\varphi(x)\right)=(y,y')$ and so $(y,y')\in \left(\iota_X\times\varphi\right)\left(\Delta_x\ri ght)$

Conversely, let $(y,y')\in\left(\varphi\times\iota_X\right)\left(\D elta_X\right)$ . Then, $(y,y')=\left(\varphi\times \iota_X\right)\left((x,x)\right)$ for some $x\in X$ but that means that $(y,y')=(x,\varphi(x))$ and so $(y,y')\in\Gamma_{\varphi}$ .

Recalling the lemma and that connectedness is invariant under continuous mappings finishes the argument. $\blacksquare$

Is that right? I feel as though I am making a stupid mistake...especially because I realized that I didn't use Hausdorffness anywhere.

**Drexel28** · March 11th 2010, 05:54 PM

Idiot moment. Of course it's true.

$\iota_X\oplus\iota_X:X\mapsto X\times X$ and $\iota_X\times\varphi:X\times X\mapsto X\times Y$ are both continuous. So, of course any topological property invariant under continuous mappings transfers from $X$ to $\Delta_X$ and $\Gamma_\varphi$

Sorry for the hiccup

**Jose27** · March 11th 2010, 06:31 PM

Originally Posted by Drexel28

I came up with the following question, and would appreciate if I could get some validation/invalidation.

If $X$ is connected Hausdorff and $\varphi:X\mapsto Y$ continuous then is $\Gamma_\varphi\subseteq X\times Y$ connected? ( $\Gamma_\varphi$ is the graph of $\varphi$ ). I think the answer is yes.

Proof: We first need a lemma

Lemma: Let $X$ be connected Hausdorff, then $\Delta_X\subseteq X\times X$ is connected ( $\Delta_X$ is the diagonal).
Proof: Clearly the map $\iota_X\oplus\iota_X:X\mapsto X\times X$ given by $x\mapsto (x,x)$ is continuous. But, it is not hard to see that $\Delta_X=\left(\iota_X\oplus \iota_X\right)\left(X\right)$ . But, connectedness is invariant under continuous mapping. So, the conclusion follows $\blacksquare$

Now, it is easy to prove that the product of two continuous maps is continuous. In particular, since $\varphi$ and the identity map $\iota_X$ are continuous we have that $\iota_X\times\varphi:X\times X\mapsto X\times Y$ is continuous. I now claim that $\Gamma_\varphi=\left(\iota_X\times\varphi\right)\l eft(\Delta_X\right)$ .

To see this, let $(y,y')\in\Gamma_\varphi$ then $(y,y')=(x,\varphi(x))$ for some $x\in X$ . Clearly then $\left(\iota_X\times\varphi\right)\left((x,x)\right )=\left(x,\varphi(x)\right)=(y,y')$ and so $(y,y')\in \left(\iota_X\times\varphi\right)\left(\Delta_x\ri ght)$

Conversely, let $(y,y')\in\left(\varphi\times\iota_X\right)\left(\D elta_X\right)$ . Then, $(y,y')=\left(\varphi\times \iota_X\right)\left((x,x)\right)$ for some $x\in X$ but that means that $(y,y')=(x,\varphi(x))$ and so $(y,y')\in\Gamma_{\varphi}$ .

Recalling the lemma and that connectedness is invariant under continuous mappings finishes the argument. $\blacksquare$

Is that right? I feel as though I am making a stupid mistake...especially because I realized that I didn't use Hausdorffness anywhere.

Please, maps to is not equal to a function...please...you're driving me crazy...

(I'm an analyst...please...quit that sh...whatever...)

**Drexel28** · March 11th 2010, 06:34 PM

Originally Posted by Jose27

Please, maps to is not equal to a function...please...you're driving me crazy...

(I'm an analyst...please...quit that sh...whatever...)

Take that!

**Jose27** · March 11th 2010, 10:44 PM

Originally Posted by Drexel28

Take that!

Okay, I guess it's my bad since I didn't specify what I was ranting about...(sorry it came off a bit insulting)

What I'm talking about is that usually one writes a function with $\rightarrow$ and the correspondence relation with $\mapsto$ (not trying to impose, but it really bugs me out...I really need to relax a bit with notation...

)

**Drexel28** · March 12th 2010, 10:51 AM

Originally Posted by Jose27

Okay, I guess it's my bad since I didn't specify what I was ranting about...(sorry it came off a bit insulting)

What I'm talking about is that usually one writes a function with $\rightarrow$ and the correspondence relation with $\mapsto$ (not trying to impose, but it really bugs me out...I really need to relax a bit with notation...

)

It's ok. I appreciate that you'd let met know.

I understand that one usually puts $\phi:G\longrightarrow G'$ and $g\mapsto g'$ . But in my naieve ignorance I got used to doing it the wrong way. It's just kind of persisted. I'll eventually wise up

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