Every Open Subset of Locally Path-Connected Space is Locally Path-Connected

Last updated Nov 1, 2022

Let $X$ be a Topological Space that is Locally Path-Connected. Then if $U \subset X$ is Open, it is Locally Path-Connected.

Since $X$ is Locally Path-Connected, there exists a Topological Basis $\mathcal{B}$ for $X$ made up of Path-Connected subsets. Because $\mathcal{B}$ is a Topological Basis, for each $U’ \subset U$ Open, there exist $\mathcal{B}{U’}$ such that $\bigcup\limits{B \in \mathcal{B}{U’}}B = U’$. Thus, taking $\bigcup\limits{U’ \subset U \text{ open}} \mathcal{B}{U’}$ forms a Topological Basis for $U$. By construction, all elements of $\mathcal{B}{U’}$ for each $U’ \subset U$ Open are Path-Connected, so $U$ is Locally Path-Connected. $\blacksquare$