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