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compact

Defines: 
compact set, compact subset
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

54D30 no label found81-00 no label found83-00 no label found82-00 no label found46L05 no label found22A22 no label found

Comments

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

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.

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.

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.

> 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.

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