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A topological space containing a countable dense subset.
A topological space is separable if it contains a countable dense subset: there is a countable set whose closure is all of . Equivalently, every nonempty open subset of meets .
For a metric space, separability says that the whole space can be approximated arbitrarily closely by countably many points. A Banach or Hilbert space is called separable when it is separable in the topology induced by its norm.
A topological space is an ordered pair where is a set and is a topology on , meaning:
Here denotes the power set of , and the members of are the open sets (whose complements are the closed sets). Many standard constructions—such as the subspace topology, product topology, and quotient topology—produce new topological spaces from existing ones.
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.