Proof That The First Countable Property is a Topological Property


The first countable property is a topological property.


Supposeandare homeomorphic so that

Letbe a homeomorphism and letbe first countable. Letand let be an open subset ofcontaining

Sinceis a homeomorphismis a point inandis an open set insuch that

is first countable hencehas a countable local baseAn open setexists such thatthenwhereis open in