Thread: Proof involving closures of functions in metric spaces

If f is a continuous mapping of a metric space into a metric space , prove that for every set . Show by an example that can be a proper subset of .

That's a lovely little problem but what have you done on it? There are, typically, many different ways of proving things like this, depending on exactly what definitions and basic ideas you use. Without seeing what you know about this situation, we don't know which to recommend for you.