Open sets in topological spaces

Question

Let $f: X \to Y$ be a function from a metric space X to another metric space Y.

Does it hold true that if a set A is open in X, then f(A) is open in Y? Or does the function has to be continuous?

Thanks for your help!

Answer 1

Even if $f$ is continuous, that doesn't have to be true. Take a constant function from $\mathbb R$ into itself, endowed with the usual metric. It is continuous, but if $A$ is non-empty open set, $A$ is mapped into a set with a single point, which is not open.

Answer 2

Well, the definition of continuity in terms of topology is take an open set V in Y, if the preimage $f^{-1}(V)$ in X is open, then $f$ is continuous. So if you know $f$ is continuous, then if $f(A)$ is open, A is. So continuity of the function doesn't help, it only tells you the other way around. Does that help?