Local Path-Connectedness is a Topological Invariant

Last updated Nov 1, 2022

Suppose $X, Y$ are Topological Spaces and $X \cong Y$. Then $X$ is Locally Path-Connected If and Only If $Y$ is.

Because the Function Inverse of a Homeomorphism is a Homeomorphism, we only need to prove the $(\Rightarrow)$ direction. Suppose $X$ is Locally Path-Connected. But this follows easily because A Function is a Homeomorphism iff it is a Bijective Continuous Open Map, Continuous Open Maps transfer Bases, and Continuous Functions Preserve Path-Connectedness. $\blacksquare$