Proof That a Subspace of a First Countable Set is First Countable

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Theorem

A subspaceof a first countable setis also first countable.

Proof

Letbe a first countable space and letbe a subspace.

LetBy hypothesisis first countable, so a countable basisin exists.

For eachdefine

Then eachis open inand form alocal subbase at

The setis countable. Henceis a first countable space.