closed subsets of a compact set are compact

Theorem 1.

Suppose $X$ is a topological space. If $K$ is a compact subset of $X$ , $C$ is a closed set in $X$ , and $C\subseteq K$ , then $C$ is a compact set in $X$ .

The below proof follows e.g. (http://planetmath.org/Eg) [3]. A proof based on the finite intersection property is given in [4].

Proof.

Let $I$ be an indexing set and $F=\{V_{\alpha}\mid\alpha\in I\}$ be an arbitrary open cover for $C$ . Since $X\setminus C$ is open, it follows that $F$ together with $X\setminus C$ is an open cover for $K$ . Thus, $K$ can be covered by a finite number of sets, say, $V_{1},\ldots,V_{N}$ from $F$ together with possibly $X\setminus C$ . Since $C\subset K$ , $V_{1},\ldots,V_{N}$ cover $C$ , and it follows that $C$ is compact. ∎

The following proof uses the finite intersection property (http://planetmath.org/ASpaceIsCompactIfAndOnlyIfTheSpaceHasTheFiniteIntersectionProperty).

Proof.

Let $I$ be an indexing set and $\{A_{\alpha}\}_{\alpha\in I}$ be a collection of $X$ -closed sets contained in $C$ such that, for any finite $J\subseteq I$ , $\displaystyle\bigcap_{\alpha\in J}A_{\alpha}$ is not empty. Recall that, for every $\alpha\in I$ , $A_{\alpha}\subseteq C\subseteq K$ . Thus, for every $\alpha\in I$ , $A_{\alpha}=K\cap A_{\alpha}$ . Therefore, $\{A_{\alpha}\}_{\alpha\in I}$ are $K$ -closed subsets of $K$ (see this page (http://planetmath.org/ClosedSetInASubspace)) such that, for any finite $J\subseteq I$ , $\displaystyle\bigcap_{\alpha\in J}A_{\alpha}$ is not empty. As $K$ is compact, $\displaystyle\bigcap_{\alpha\in I}A_{\alpha}$ is not empty (again, by this result (http://planetmath.org/ASpaceIsCompactIfAndOnlyIfTheSpaceHasTheFiniteIntersectionProperty)). This proves the claim. ∎

References

1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
2 S. Lang, Analysis II, Addison-Wesley Publishing Company Inc., 1969.
3 G.J. Jameson, Topology and Normed Spaces, Chapman and Hall, 1974.
4 I.M. Singer, J.A. Thorpe, Lecture Notes on Elementary Topology and Geometry, Springer-Verlag, 1967.

Title	closed subsets of a compact set are compact
Canonical name	ClosedSubsetsOfACompactSetAreCompact
Date of creation	2013-03-22 13:55:56
Last modified on	2013-03-22 13:55:56
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	16
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 54D30
Related topic	AClosedSetInACompactSpaceIsCompact
Related topic	ACompactSetInAHausdorffSpaceIsClosed