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?

I think of a "space" as the conceptually smallest place in which a given abstraction makes sense. For example, in a metric space, we have distilled the notion of distance. In a topological space, we are in the minimal setting for continuity.  
