# Homework Help: Function continuity in metric spaces

1. Apr 10, 2010

### complexnumber

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

Let $$(X,d_X)$$ and $$(Y,d_Y)$$ be metric spaces and let $$f: X \to Y$$.

2. Relevant equations

Prove that the following statements are equivalent:

1. $$f$$ is continuous on $$X$$,
2. $$\overline{f^{-1}(B)} \subseteq f^{-1}(\overline{B})$$ for all subsets $$B \subseteq Y$$

3. The attempt at a solution

I an prove that (1) leads to (2) but don't know how to show (2) leads to (1). Can you give me some hint?

2. Apr 11, 2010

### snipez90

A function f: X -> Y between topological spaces is continuous if and only if the inverse image of every closed set in Y is a closed set in X.

Let F be an arbitrary closed set in Y. Then F = cl(F), where cl(F) is the closure of F. Now apply (2) with F in place of B.

3. Apr 21, 2010

### complexnumber

Thanks, I didn't realize I should use this theorem and was trying to prove it using any point x and its neighborhood approach.