(adsbygoogle = window.adsbygoogle || []).push({}); 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 [tex]f[/tex] between Hausdorff spaces is continuous

if and only if the sequence [tex]f(x_0), f(x_1), \dots[/tex] converges to [tex]f(x)[/tex]

for every sequence [itex]x_0, x_1, \dots[/itex] converging to [itex]x[/itex].

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Hausdorff spaces & Continuous Mappings via Convergence.

**Physics Forums | Science Articles, Homework Help, Discussion**