General Topology/Countability, density

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Definition (dense):

Let be a topological space and let be a subset. is called dense if and only if .

Definition (first-countable):

Let be a topological space. is called first-countable iff for all the neighbourhood filter has a countable basis.

Definition (second-countable):

Let be a topological space. is called second-countable iff the topology of has a countable basis.

Since subsets of countable sets are countable and the open neighbourhoods generate , second-countability implies first-countability.

Definition ():

separable space

Proposition ():

second-countable spaces are separable

(On the condition of the axiom of countable choice.)

Proof: Let be a basis of the topology of and choose . Then is countable and dense.

Proposition ():

subspace of second-countable space is second-countable

Proof: Any countable basis of the topology of induces a countable basis of the subspace topology on .

Proposition ():

continuous function into Hausdorff space is uniquely determined by dense subspace

Proof: Let be arbitrary, and let be any neighbourhood of . By continuity of we may find a neighbourhood of that is mapped completely into . Analogously, whenever is a neighbourhood of , we find a neighbourhood mapping completely into . Then is mapped completely into , so that for any open neighbourhoods of and of . If , then for suitable as above by the Hausdorff condition, a contradiction to . Hence, . Since was arbitrary, we conclude.