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):

Let be a topological space. is called separable if and only if there exists a countable set which is dense in .

Proposition (second-countable spaces are separable):

Let be a second-countable space. Then is 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):

Let be a second-countable space, and a subset. Turn into a topological space using the subspace topology. is then a second-countable space.

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):

Let be topological spaces, where is Hausdorff. Let be a dense subspace, and suppose is continuous. Whenever are continuous functions such that , then .

Proof: Let be arbitrary, and let be any neighbourhood of . By continuity of 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.