Homeomorphisms transfer Bases

Last updated Nov 1, 2022

Let $X, Y$ be Topological Spaces and let $X$ have Topological Basis $\mathcal{B}$. Suppose there exists Homeomorphism $\varphi: X \to Y$. Then $\varphi(\mathcal{B}) := {\varphi(B) \subset Y : B \in \mathcal{B}}$ is a Topological Basis for $Y$.

A Function is a Homeomorphism iff it is a Bijective Continuous Open Map, and Continuous Open Maps transfer Bases.