Locally Euclidean Subspace of Euclidean Space is Manifold

Theorem

Let $\R^n$ be an Euclidean space for $n \in \N$.

Let $M = \struct{H, \tau_H}$ be a subspace of $\R^n$, where $H \subseteq \R^n$.

Let $M$ be locally Euclidean of dimension $d$.


Then $M$ is a $d$-manifold.


Proof

From Metric Space is Hausdorff, $\R^n$ is a Hausdorff space.

From Subspace of Hausdorff Space is Hausdorff, $M$ is a Hausdorff space.

From Euclidean Space is Second-Countable, $\R^n$ is second-countable.

From Second-Countability is Hereditary, $M$ is second-countable.

Therefore, $M$ is a $d$-manifold.

$\blacksquare$


Sources

  • 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: New Spaces From Old: Subspaces. Topological Embeddings
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