Surjective Embedding is Homeomorphism

Theorem

Let $T_1 = \struct{S_1, \tau_1}$ and $T_2 = \struct{S_2, \tau_2}$ be topological spaces.

Let $f: S_1 \to S_2$ be a surjective embedding.

Then $f$ is a homeomorphism.

Proof

Let $f {\restriction_{S_1 \times f\sqbrk {S_1} }}$ be the restriction of $f$ to its image.

By definition of surjection, $f \sqbrk {S_1} = S_2$.

Therefore, $f {\restriction_{S_1 \times f\sqbrk {S_1} }} = f$

By definition of embedding, $f {\restriction_{S_1 \times f\sqbrk {S_1} }}$ of $f$ to its image is a homeomorphism.

Therefore, $f$ is a homeomorphism.

$\blacksquare$

Sources

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