Connected Hausdorff spaces and graphs
I came up with the following question, and would appreciate if I could get some validation/invalidation.
If 
is connected Hausdorff and 
continuous then is 
connected? ( 
is the graph of 
). I think the answer is yes.
Proof: We first need a lemma
Lemma: Let be connected Hausdorff, then 
is connected ( 
is the diagonal).
Proof: Clearly the map 
given by )
is continuous. But, it is not hard to see that \left(X\right))
. But, connectedness is invariant under continuous mapping. So, the conclusion follows 
Now, it is easy to prove that the product of two continuous maps is continuous. In particular, since and the identity map 
are continuous we have that 
is continuous. I now claim that \l eft(\Delta_X\right))
.
To see this, let \in\Gamma_\varphi)
then =(x,\varphi(x)))
for some 
. Clearly then \left((x,x)\right )=\left(x,\varphi(x)\right)=(y,y'))
and so \in \left(\iota_X\times\varphi\right)\left(\Delta_x\ri ght))
Conversely, let \in\left(\varphi\times\iota_X\right)\left(\D elta_X\right))
. Then, =\left(\varphi\times \iota_X\right)\left((x,x)\right))
for some but that means that and so \in\Gamma_{\varphi})
.
Recalling the lemma and that connectedness is invariant under continuous mappings finishes the argument.
Is that right? I feel as though I am making a stupid mistake...especially because I realized that I didn't use Hausdorffness anywhere.
