Locally Path-Connected Spaces are Path-Connected iff they are Connected

Last updated Nov 1, 2022

Let $X$ be a Topological Space that is Locally Path-Connected. Then $X$ is Path-Connected If and Only If $X$ is Connected

$(\Rightarrow)$: This direction follows from the general case: Path-Connected implies Connected.

$(\Leftarrow)$: If $X$ is Connected, then it has a single Connected Component that is $X$ itself. Since Path-Connected Components are equal to the Connected Components in a Locally Path-Connected Space, $X$ is a Path-Connected Component. But this just means $X$ is Path-Connected. $\blacksquare$