As I've understood up to now, Pure Topological spaces and Vector spaces are the most general form of spaces that we may think of in math. Now it seems to me that whenever i see the word space in math, I should be certain that the space is either a topological space or a vector space or both. Now my question is that am i right? For example are all the spaces we may confront in non-Euclidean geometry such as hyperbolic or Minkowski at the heart Topological or a Vector space or both? Moreover as an extra question are all spaces used in geometry ( Euclidean and non-Euclidean) even metric, because i believe that distance is the characteristic feature of all geometric spaces. Just even a short hint would be appreciated.
You can define your own kind of spaces if you want.
There are other kinds of spaces, for example measure and probability spaces.