Topological Manifold is Locally Path-Connected
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Theorem
Let $M$ be a topological manifold.
Then $M$ is a locally path-connected space.
Proof
By definition of manifold:
- $M$ is a locally Euclidean space
The result follows from Locally Euclidean Space is Locally Path-Connected
$\blacksquare$
Sources
- 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.): Chapter $4.$ Connectedness and Compactness, Proposition $4.23$