Compact to Hausdorff implies closed

Statement

Any continuous map from a compact space to a Hausdorff space is a closed map i.e. the image of any closed set is closed.

Applications

• Any surjective continuous map from a compact space to a Hausdorff space is a quotient map
• Any continuous injective map from a compact space to a Hausdorff space is a subspace embedding

Proof

We need to show that given any closed subset of the compact space, its image is closed in the Hausdorff space.

We reason in the following steps:

• The closed subset of the compact space, is itself compact in the subspace topology. This is because any closed subset of a compact space is compact. For full proof, refer: Compactness is weakly hereditary
• Thus, the image of this subset, in the target space, is also compact in the subspace topology. This follows from the fact that an image of a compact space under a continuous map is also compact. Further information: Compactness is continuous image-closed
• The image of this subset is a compact subset of a Hausdorff space, hence is closed in the Hausdorff space. This follows from the fact that any compact subset of a Hausdorff space is closed. Further information: Hausdorff implies KC