Proof That Every Subspace of a Second Countable Space is Second Countable

Theorem

Every subspaceof a second countable spaceis second countable.

Proof

Letbe a countable base for the second countable space

For any subspaceis a countable base forHenceis second countable.

Also, any second countable space is separable. A spacecontains a countable dense subset.

Letbe a second countable space with a countable baseForselectThe setis a countable dense subset ofhenceis separable.