Difference between a "topology" and a "space"? What do we mean when we talk about a topological _space_ or a metric _space_? I see some people calling metric topologies metric spaces and I wonder if there is some synonymity between a topology and a space? What is it that the word means, and if there are multiple meanings how can one distinguish them?

In mathematics, you usually call a set (a collection of objects) with some additional structures a space. So for example, a set with a certain distance function is called a metric space, and a set with certain subsets defined to be open is called a topological space. (of course in these two examples the distance function and open sets have to satisfy certain axioms.) There are tons of other spaces too, vector spaces etc.

The reason some people refer to a topological space as as having a metric topology, is because a metric space is a specific example of a topological space. hope that helps.