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

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 $X$, a Hausdorff space $Y$, a continuous map $f:X\to Y$.

To prove: For any closed subset $A$ of $X$, $f(A)$ is a closed subset of $Y$.

Proof:

1. $A$ is compact under the subspace topology: This follows from the given datum that $X$ is compact and fact (1).
2. $f(A)$ is compact under the subspace topology in $Y$: First, note that the map $f|_{A}:A\to Y$ is continuous, because it is the composite of the inclusion of $A$ in $X$ with the map $f$, both of which are continuous. Thus, by fact (2), $f(A)$ is compact with the subspace topology from $Y$.
3. $f(A)$ is closed in $Y$: This follows from the previous step, the given datum that $Y$ is Hausdorff, and fact (3).