# Hausdorff spaces & Continuous Mappings via Convergence.

1. Oct 22, 2007

### strangerep

1. The problem statement, all variables and given/known data

A kind person is helping me to self-study some aspects of topology,
continuity, etc. He posed the following exercise for me, which I
can't do, but he doesn't have time to write up the full solution.

Ex: Show that a mapping $$f$$ between Hausdorff spaces is continuous
if and only if the sequence $$f(x_0), f(x_1), \dots$$ converges to $$f(x)$$
for every sequence $x_0, x_1, \dots$ converging to $x$.

2. Relevant equations

3. The attempt at a solution

To show f is continuous, one must show that every open set in the range of f(x) must
have an inverse image which is also an open set. I also know the definition
of a Hausdorff space as a topological space in which any two points are
contained in disjoint open sets. But that's about as far as I get. Does this
theorem have a name that I could look up? Or can anyone give me some
more hints about how to progress the proof?

TIA.

2. Oct 23, 2007

### EnumaElish

From http://en.wikipedia.org/wiki/Hausdorff_spaces#Properties

Which tells me you can start from the reals (as a special case) then generalize.