CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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An injective immersion from a compact manifold is an embedding

Statement

Let $M$ be compact and let $N$ be Hausdorff. Every injective smooth immersion $F : M \to N$ is a smooth embedding.

Facts & Assumptions

Given: A compact manifold $M$ , a Hausdorff manifold $N$ , and an injective smooth immersion $F : M \to N$ .

[F1]

A smooth embedding is an injective immersion and a homeomorphism onto its image with the subspace topology (Smooth embeddings).

[L1]

Smooth maps are continuous (Smooth maps are continuous).

[L2]

Closed subsets of compact spaces are compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right).

[L3]

Compact subsets of Hausdorff spaces are closed (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not).

[L4]

Hausdorffness is hereditary to subspaces ( $T_{0}$ , $T_{1}$ , and Hausdorffness are hereditary).

Proof

technique · direct

1.1

By [L1], $F$ is continuous. Since it is injective, the corestriction $f_{0} : M \to F (M)$ is a continuous bijection. By [L4], the subspace $F (M) \subseteq N$ is Hausdorff.

L1L4given

2.1

Let $C \subseteq M$ be closed. Since $M$ is compact, [L2] makes $C$ compact. To show that $f_{0} [C]$ is compact in $F (M)$ , let $U$ be an open cover of $f_{0} [C]$ in the subspace $F (M)$ ; then ${f_{0}^{- 1} (U) : U \in U}$ is an open cover of $C$ , so finitely many members cover $C$ . Applying $f_{0}$ back shows that the same finite subfamily covers $f_{0} [C]$ . Thus $f_{0} [C]$ is compact, hence closed in the Hausdorff space $F (M)$ by [L3].

L1L2L3step 1.1

3.1

Step 2.1 shows that $f_{0}$ is a closed bijection. Therefore for every open set $O \subseteq M$ , the complement $M ∖ O$ is closed and $f_{0} [O] = F (M) ∖ f_{0} [M ∖ O]$ is open in $F (M)$ . So $f_{0}$ is an open bijection, hence a homeomorphism. Since $F$ is an injective immersion by hypothesis, [F1] now gives that $F$ is a smooth embedding.

F1step 1.1step 2.1∎

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Sources

John M. Lee, Introduction to Smooth Manifolds, Embeddings (standard reference, not scraped)
Will J. Merry, Differential Geometry, Proposition 6.3 (standard reference, not scraped)