## You are here

Homeclosed set in a compact space is compact

## Primary tabs

# closed set in a compact space is compact

*Proof.* Let $A$ be a closed set in a compact space $X$.
To show that $A$ is compact, we show that an arbitrary open cover has
a finite subcover. For this purpose, suppose
$\{U_{i}\}_{{i\in I}}$ be an arbitrary open cover for $A$.
Since $A$ is closed, the complement of $A$,
which we denote by $A^{c}$, is open.
Hence
$A^{c}$ and $\{U_{i}\}_{{i\in I}}$ together form an open cover for $X$.
Since $X$ is compact, this cover has a finite subcover that
covers $X$. Let $D$ be this subcover.
Either $A^{c}$ is part of $D$ or $A^{c}$ is not.
In any case, $D\backslash\{A^{c}\}$ is a finite open cover
for $A$, and $D\backslash\{A^{c}\}$
is a subcover of $\{U_{i}\}_{{i\in I}}$. The claim follows. $\Box$

Related:

ClosedSubsetsOfACompactSetAreCompact

Major Section:

Reference

Type of Math Object:

Proof

Parent:

## Mathematics Subject Classification

54D30*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

## Comments

## mention article 4691

article 4691 called "closed subsets of a compact set are compact" deals with almost the same subject matter an should be mentioned.