nLab closed subspaces of compact spaces are compact




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

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Let (X,τ)(X,\tau) be a compact topological space, and let YXY \subset X be a closed topological subspace. Then also YY is compact.


Let {V iY} iI\{V_i \subset Y\}_{i \in I} be an open cover of YY. We need to show that this has a finite sub-cover.

By definition of the subspace topology, there exist open subsets U iU_i of XX with

V i=U iY. V_i = U_i \cap Y \,.

By the assumption that YY is closed, the complement XYX \setminus Y is an open subset of XX, and therefore

{XYX}{U iX} iI \{ X \setminus Y \subset X\} \cup \{ U_i \subset X \}_{i \in I}

is an open cover of XX. Now by the assumption that XX is compact, this latter cover has a finite subcover, hence there exists a finite subset JIJ \subset I such that

{XYX}{U iX} iJI \{ X \setminus Y \subset X\} \cup \{ U_i \subset X \}_{i \in J \subset I}

is still an open cover of XX, hence in particular intersects to a finite open cover of YY. But since Y(XY)=Y \cap ( X \setminus Y ) = \empty, it follows that indeed

{V iY} iJI \{V_i \subset Y\}_{i \in J \subset I}

is a cover of YY, and is indeed a finite subcover of the original one.

Last revised on November 24, 2022 at 11:24:59. See the history of this page for a list of all contributions to it.