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# compact

A topological space $X$ is compact if, for every collection $\{U_{i}\}_{{i\in I}}$ of open sets in $X$ whose union is $X$, there exists a finite subcollection $\{U_{{i_{j}}}\}_{{j=1}}^{n}$ whose union is also $X$.

A subset $Y$ of a topological space $X$ is said to be compact if $Y$ with its subspace topology is a compact topological space.

Note: Some authors require that a compact topological space be Hausdorff as well, and use the term quasi-compact to refer to a non-Hausdorff compact space. The modern convention seems to be to use compact in the sense given here, but the old definition is still occasionally encountered (particularly in the French school).

Defines:

compact set, compact subset

Related:

QuasiCompact, LocallyCompact, HeineBorelTheorem, TychonoffsTheorem, Compactification, SequentiallyCompact, Lindelof, NoetherianTopologicalSpace

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

54D30*no label found*81-00

*no label found*83-00

*no label found*82-00

*no label found*46L05

*no label found*22A22

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## Attached Articles

examples of compact spaces by yark

properties of compact spaces by rspuzio

$Y$ is compact if and only if every open cover of $Y$ has a finite subcover by mathcam

a space is compact iff any family of closed sets having fip has non-empty intersection by CWoo

a compact set in a Hausdorff space is closed by mathcam

compactness is preserved under a continuous map by yark

closed subsets of a compact set are compact by Wkbj79

continuous image of a compact set is compact by Wkbj79

properties of compact spaces by rspuzio

$Y$ is compact if and only if every open cover of $Y$ has a finite subcover by mathcam

a space is compact iff any family of closed sets having fip has non-empty intersection by CWoo

a compact set in a Hausdorff space is closed by mathcam

compactness is preserved under a continuous map by yark

closed subsets of a compact set are compact by Wkbj79

continuous image of a compact set is compact by Wkbj79

## Comments

## Historical note and terminology

Maybe it would be a good idea to mention that

in some european (especially russian) literature

what is known as compact used to be referred

to as "bicompact", and "compact" referred to

what is now known as countably compact.

Also, I've seen references to "bicompact" meaning

compact and Hausdorff.

My sources are

L. S. Pontriagin. _Continuous Groups_. National Publishing House for Technico-Theoretical Literature, 2nd edition, Moscow: 1954.

http://cm.bell-labs.com/who/will/CAARMS5/williams.pdf

## adopt one definition?

Since some authors require compact spaces to be Hausdorff and others don't, it is apropriate to mention this fact in the article. But I wonder if PlanetMath should adopt one convention, so that everybody's automatic links to the compact article will give the proper definition. Otherwise, everybody has to state explicitly in every article what they mean when they say compact.

## Re: adopt one definition?

I think it's clear by now that the non-Hausdorff definition of compact has won out in modern literature. Therefore planetmath should adopt the convention that compact spaces are not necessarily Hausdorff. Since people seem to be doing this anyway, I don't think anything else needs to be done.

## Re: adopt one definition?

But shouldn't we make this convention explicit? When I learned topology, compact spaces were still Hausdorff, so the first thing I do when I see the word "compact" is to check the definition used. But in PlanetMath right now, that wouldn't help me, since the compactness entry doesn't explicitly say which definition PlanetMath uses.

## Re: adopt one definition?

> the compactness entry doesn't explicitly say which

> definition PlanetMath uses.

Even if you think the entry itself is not explicit enough, the contents of this discussion thread (which is displayed every time the entry is displayed) should make it quite clear that the intended convention on PlanetMath is that compact spaces are not always Hausdorff.

I do not want to make it an absolute declaration that PlanetMath will always use the modern convention because, in PlanetMath as in real life, one occasionally encounters those who use the old convention. The fact that a PlanetMath reader must suffer some ambiguity about compactness is a good thing, because that's the way it is in real life.