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 $C\mathrm{\subseteq}K$, 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, ${V}_{1},\mathrm{\dots},{V}_{N}$ from $F$ together with possibly $X\setminus C$. Since $C\subset K$, ${V}_{1},\mathrm{\dots},{V}_{N}$ 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 $J\subseteq I$, $\bigcap _{\alpha \in J}}{A}_{\alpha$ 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 $J\subseteq I$, $\bigcap _{\alpha \in J}}{A}_{\alpha$ is not empty. As $K$ is compact, $\bigcap _{\alpha \in I}}{A}_{\alpha$ 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 |