Topological Manifolds are Locally Path-Connected

Last updated Nov 1, 2022

Let $M$ be a Topological Manifold of Manifold Dimension $n$. $M$ is Locally Path-Connected.

Consider a Coordinate Ball $B \subset M$. Because

• The Restriction of Homeomorphism is a Homeomorphism in the Subspace Topology
• Path-Connectedness is a Topological Invariant

$B$ is Path-Connected. Because Coordinate Balls form a Basis for Manifolds, we have that $M$ is Locally Path-Connected. $\blacksquare$