Compact to Hausdorff implies closed

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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

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).