Topological Manifold with Boundary is Topological Manifold iff Boundary is Empty

Theorem

Let $M$ be a topological manifold with boundary.

Then, $M$ is a topological manifold if and only if its boundary $\partial M$ is empty.

Proof

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Sources

• 2013: John M. Lee: Introduction to Smooth Manifolds (2nd ed.): Chapter $1$: Smooth Manifolds: $\S$ Manifolds with Boundary