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?

A metric space is a space in which the notion of distance is defined: there is a distance function that exists in that space, and it is true for all points in that space. For example, the distance formula between two points in Euclidean space. 
A topological space is a space in which a notion of "closeness" or "nearness" is understood as existing between some points.
