NetNews Usenet Archive 1992 #27/NN_1992_27.iso/spool/sci/math/15344

Internet Message Format | 1992-11-21 | 2.6 KB

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From: [e-mail] (John Mount)
Newsgroups: sci.math
Subject: Re: compactness ?
Message-ID: <[e-mail]>
Date: 21 Nov 92 18:08:14 GMT
Article-I.D.: cs.By2vpr.EpJ.2
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In article <[e-mail]>, [e-mail] (test user for max) writes:
|> I am trying to teach myself analysis; i'm stuck on the notion of compactness.
|> One book's definition:
|>   A subset S of a metric space E is compact if whenever S is
|>   contained in the union of a collection of open subsets of E,
|>   then S is contained in the union of a finite number of these
|>   open subsets.

You should read the above definition as:

  A subset S of a metric space E is compact if for EVERY collection of
  open subsets of E such that the union of the members of E contains S
  there is a finite collection E' contained in E such that S is contained
  the union of the members of E'.

|> The book (M.Rosenlicht,Intro.to Analysis) then gives an example of
|> a non-compact set- the open interval (0,1), which is contained in the
|> union of sets (1/n,1) but not contained in any finite number of these.
|>
|> ? It seems to me that (0,1) is contained in the collection of
|> open sets { (0,0.6) (0.5,1) (0,1) } and this has finite subsets that
|> contain (0,1), so (0,1) should be compact.  Does the definition mean that
|> S must be contained in any possible collection, as opposed
|> to just one possible collection being sufficient?  If so, then
|> I would intuitively think that a (1/n,..) type construction
|> would prevent any open set from being compact.
|> Is a 'collection of sets' a stricter notion than I'm assuming?
|> (Is 'contained in' the same as 'covered by'?)

The closed interval [0,1] in R with the usual topology is compact.  I
won't prove it here- but you will notice that this fact isn't
contradicted by the collection E = {(1/n,2),n=1,2....} example because
while no finite subcollection of E covers [0,1] you will notice that E
doesn't cover [0,1] either (0 isn't a member of any member of E- so
fails to be covered).

--
--- It is kind of strange being in CS theory, given computers really do exist.
John Mount: [e-mail]               [phone]
School of Computer Science, Carnegie Mellon University,
5000 Forbes Ave., Pittsburgh PA 15213-3891