Compact to Hausdorff implies closed

From Topospaces
Jump to: navigation, search


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.


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.


  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).