# Compact to Hausdorff implies closed

From Topospaces

## Contents

## 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

## Facts used

- Compactness is weakly hereditary: Any closed subset of a compact space is compact in the subspace topology.
- Compactness is continuous image-closed: The image of a compact space under a continuous map is a compact space.
- 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**:

- is compact under the subspace topology: This follows from the given datum that is compact and fact (1).
- 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 .
- is closed in : This follows from the previous step, the given datum that is Hausdorff, and fact (3).