Quote by michonamona
Thank you for your replies.
Would it be safe to make the following generalization?
topological space>metric space>euclidean space
This means that every euclidean space is a metric space and every metric space is a topological space. By transitivity, every euclidean space is a topological space.

yes. certain spatial properties of euclidean space are abstracted to get the notion of a topological space.
metric spaces are inbetween the two, they are a special kind of topological space, but there are several possible metrics on a given set, including R^n. of these, only one is the standard euclidean metric on R^n: d(x,y) = √(<x  y,x  y>).