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

. 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

.

To see this, let

then

for some

. Clearly then

and so

Conversely, let

. Then,

for some

but that means that

and so

.

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.