Re: Compact vs Quasicompact

From: Marc Olschok (sa796ol_at_l1-hrz.uni-duisburg.de)
Date: 10/08/04 

Date: Fri, 8 Oct 2004 15:39:36 +0000 (UTC)

Jose Capco <nospam@nospam.org> wrote:
> Dear NG,
>
> Just a while ago I was discussing with my prof. about something in a
> book. We came to term "Quasicompactness", which I thought as kinda wierd
> coz it is just the definition of compactness as I knew since ages
> ie. every open covering has a finite subcovering (that was the
> definition of quasicompactness in the book .. Hartshorne again *sweat*)
> Well my prof. just told me that I had the wrong definition, which just
> shoked me. He said that compactness also includes Hausdorfness. I was
> totally, uhm speechless... I don't know whether I have been having the
> wrong definition of compactness all this while or was it that my
> professor is just having a nonstandard definition of compactness. As far
> as I know, or at least since the time I attended my first topology
> class.. compactness is just what here my professor calls quasicompact.
> Would anyone please enlighten me.

In my view, the professors comment that your definition is just "wrong"
is a bit harsh; perhaps he is not that familiar with the topological
literature?

Some people prefer the pair "compact" and "compact Hausdorff", others
prefer the pair "quasicompact" and "compact". The former seems to
be universal in english texts, while the latter can still be found
in Bourbaki and older "continental" topological texts.

Perhaps you have already met the similar situation with
"ring" and "ring with 1" vs. "ring without one" and "ring".

I guess (and this is purely guessing, I do not have information on this)
that the notion of compactness was first introduced in times when
only Hausdorff topological spaces were considered; when non-Hausdorff
spaces found their way into mainstream topology it was not clear, how
the notion of compactness should be transferred to the new setting.

Personally, I prefer your version, which also seems to become more
and more dominant. So, time is on your side.

However, as long as a text makes clear, which terms are used, it should
not pose any problems to accept the (perhaps old-fashioned :-) conventions
in algebraic geometry.

Marc

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