nLab
map from compact space to Hausdorff space

Maps from compact spaces to Hausdorff spaces

Properties

There is a classic theorem about maps from compact spaces to Hausdorff spaces which states:

Theorem

Let CC be a compact space and HH be a Hausdorff space. Let f:CHf\colon C \to H be a continuous bijection. Then ff is a homeomorphism.

An immediate corollary is that the two spaces involved are both compact Hausdorff spaces.

The idea that this captures is that Hausdorffness is about having “enough” open sets whilst compactness is about having “not too many”. Thus a compact Hausdorff space has both “enough” and “not too many”. This theorem says that both conditions are at their limit: if we try to have more open sets, we lose compactness. If we try to have fewer open sets, we lose Hausdorffness.

The above theorem is a special case of a slightly more general result.

Theorem

Let CC be a compact space and HH be a Hausdorff space. Let f:CHf\colon C \to H be a continuous map. Then ff is closed and proper.

To prove this, we need to show that a closed subset of CC is taken to a closed subset of HH and that the preimage of a compact subset of HH is compact in CC. Both follow from the fact that closed subsets of a compact set are compact and that compact subsets of a Hausdorff space are closed.

For the first, let DCD \subseteq C be closed. As it is a closed subset of a compact space, it is compact. Since ff is continuous, f(D)f(D) is a compact subset of HH. Thus as HH is a Hausdorff space, f(D)f(D) is closed.

For the second, let GHG \subseteq H be compact. As it is a compact subset of a Hausdorff space, it is closed. Thus as ff is continuous, f 1(G)f^{-1}(G) is closed in CC. Hence as CC is compact, f 1(G)f^{-1}(G) is compact.

Revised on September 19, 2011 17:20:44 by Toby Bartels (71.31.209.1)