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compact (Definition)

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).



"compact" is owned by djao. [ full author list (2) ]
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See Also: quasi-compact, locally compact, Heine-Borel theorem, Tychonoff's theorem, compactification, sequentially compact, Lindelöf space, Noetherian topological space

Also defines:  compact set, compact subset

Attachments:
examples of compact spaces (Example) by yark
properties of compact spaces (Result) by rspuzio
$Y$ is compact if and only if every open cover of $Y$ has a finite subcover (Theorem) by mathcam
a space is compact if and only if the space has the finite intersection property (Theorem) by CWoo
a compact set in a Hausdorff space is closed (Theorem) by mathcam
compactness is preserved under a continuous map (Theorem) by yark
closed subsets of a compact set are compact (Theorem) by Wkbj79
continuous image of a compact set is compact (Theorem) by Wkbj79
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Cross-references: quasi-compact, Hausdorff, subspace topology, subset, finite, union, open sets, collection, topological space
There are 392 references to this entry.

This is version 6 of compact, born on 2001-10-25, modified 2004-03-28.
Object id is 503, canonical name is Compact.
Accessed 47771 times total.

Classification:
AMS MSC: 54D30 (General topology :: Fairly general properties :: Compactness)
Pending Errata and Addenda
None.
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forum policy
adopt one definition? by AxelBoldt on 2002-06-06 22:18:26
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.
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Historical note and terminology by igor on 2002-05-22 02:49:45
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
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