TITLE: What is the difference between topological and metric spaces?
QUESTION [27 upvotes]: What is the difference between a topological and a metric space?

REPLY [2 votes]: If metric space is interpreted generally enough, then there is no difference between topology and metric spaces theory (with continuous mappings). Building on ideas of Kopperman, Flagg proved in this article that with a suitable axiomatization, that of value quantales, every topological space is metrizable. This gives rise to a precise equivalence between the category of topological spaces and the category of generalized metric spaces, presented here (alg. univ.).