Injection from compact to Hausdorff implies embedding

Statement

Any injective continuous map from a compact space to a Hausdorff space is an embedding; in other words, it is a homeomorphism to its image, when the image is given the subspace topology.

Proof

Proof idea

We use two facts:

• any continuous map from a compact space to a Hausdorff space is closed
• any closed injective map is a subspace embedding