Proof That Every Open Cover of a Subset of a Second Countable Space is Reducible to a Second Countable Cover
Ifis a subset of a second countable spacethen overy open cover ofis reducible to a finite cover.
Letbe a countable base forand let be an open cover ofso that
For eachexists such that
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