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.