open subspaces of compact Hausdorff spaces are locally compact



topology (point-set topology, point-free topology)

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(locally compact topological space)

A topological space XX is called locally compact if for every point xXx \in X and every open neighbourhood U x{x}U_x \supset \{x\} there exists a smaller open neighbourhood V xU xV_x \subset U_x whose topological closure is compact and still contained in UU:

{x}V xCl(V x)compactU x. \{x\} \subset V_x \subset \underset{\text{compact}}{Cl(V_x)} \subset U_x \,.

Statement and proof


(open subspaces of compact Hausdorff spaces are locally compact)

Every open topological subspace XopenKX \underset{\text{open}}{\subset} K of a compact Hausdorff space KK is a locally compact topological space.

In particular every compact Hausdorff space itself is locally compact.


Let xXx \in X be a point and let U xXU_x \subset X an open neighbourhood. We need to produce a small open neighbourhood whose closure is compact and still contained in U xU_x.

By the nature of the subspace topology there exists an open subset V xKV_x\subset K such that U x=XV xU_x = X \cap V_x. Since XX is assumed to be open, it follows that UU is also open as a subset of KK. Since compact Hausdorff spaces are normal it follows (by this prop.) that there exists a smaller open neighbourhood W xKW_x \subset K whose topological closure is still contained in U xU_x, and since closed subspaces of compact spaces are compact:

{x}W xCl(W x)cptV xK. \{x\} \subset W_x \subset \underset{\text{cpt}}{Cl(W_x)} \subset V_x \subset K \,.

The intersection of this situation with XX is the required smaller compact neighbourhood Cl(W x)XCl(W_x) \cap X:

{x}W xXCl(W x)cptXU xX. \{x\} \subset W_x \cap X \subset \underset{\text{cpt}}{Cl(W_x)} \cap X \subset U_x \subset X \,.

Conversely, every locally compact Haudorff space XX arises as in prop. 1, since it may be considered an open subspace in its one-point compactification X{}X \sqcup \{\infty\}. See there this example.

Revised on June 3, 2017 04:56:29 by Urs Schreiber (