topological space

A topological space is a type of generalized mathematical space in which the idea of closeness, or limits, is described in terms of relationships between sets rather than in terms of distance. Every topological space consists of: (1) a set of points; (2) a class of subsets defined axiomatically as open sets; and (3) the set operations of union and intersection.

If a topological space is metrizable it means that there exists a metric compatible with the topology. To say that a topological space is metrizable is to treat it as a metric space, but without distinguishing any specific or preferred distance function.