# 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.

## Facts used

1. Compactness is weakly hereditary: Any closed subset of a compact space is compact in the subspace topology.
2. Compactness is continuous image-closed: The image of a compact space under a continuous map is a compact space.
3. Hausdorff implies KC: Any compact subset of a Hausdorff space is closed.

## Proof

Given: A compact space , a Hausdorff space , a continuous map .

To prove: For any closed subset  of ,  is a closed subset of .

Proof:

1.  is compact under the subspace topology: This follows from the given datum that  is compact and fact (1).
2.  is compact under the subspace topology in : First, note that the map  is continuous, because it is the composite of the inclusion of  in  with the map , both of which are continuous. Thus, by fact (2),  is compact with the subspace topology from .
3.  is closed in : This follows from the previous step, the given datum that  is Hausdorff, and fact (3).