Do topological theorems hold in an arbitrary metric space?

Question

This photo from this Wikipedia page suggests that metric spaces can be thought of loosely as a subset of topological spaces. Hence, are there any theorems or results that hold for arbitrary topological spaces but do not hold in arbitrary metric spaces?

Answer 1

No, since every metric space is a topological space. That's like asking if there are theorems about polygons which are not valid for triangles.

Answer 2

A metric space is a topological space, hence all theorems or results that hold for arbitrary topological spaces hold in arbitrary metric spaces.