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