Proof That Every Open Cover of a Subset of a Second Countable Space is Reducible to a Second Countable Cover

Theorem

Ifis a subset of a second countable spacethen overy open cover ofis reducible to a finite cover.

Proof

Letbe a countable base forand let be an open cover ofso that

For eachexists such that

Thus

The family of setsis a subset ofhence hence it is countable sowhereis a subset ofFor eachwe can choose such that

Henceandis a countable subcover of