# 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 ClosedSubsetsOfACompactSetAreCompact 2013-03-22 13:55:56 2013-03-22 13:55:56 Wkbj79 (1863) Wkbj79 (1863) 16 Wkbj79 (1863) Theorem msc 54D30 AClosedSetInACompactSpaceIsCompact ACompactSetInAHausdorffSpaceIsClosed