Direct proof of compactness
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From: Victor Porton on 31 Mar 2010 16:53 (Oh, I can do this work myself but why not ask sci.math?) I need this for my research in which I attempt to generalize the notion of compact topological space. In Bourbaki there is a proof that A space X is (quasi-)compact (that is having every filter in X convergent) is equivalent to Every collection of closed sets in X whose intersection is empty includes finite subcollection with empty intersection. I want DIRECT proof that (Quasi-)compactness (that is having every filter in X convergent) is equivalent to Every cover of the space X includes a finite cover of X.
From: Dave L. Renfro on 31 Mar 2010 17:08 Victor Porton wrote (in part): > I want DIRECT proof that > > (Quasi-)compactness (that is having every filter > in X convergent) > > is equivalent to > > Every cover of the space X includes a finite cover of X. I don't know if a direct proof of what you want is in either of the following, but they both seem worth looking at: http://www.efnet-math.org/~david/mathematics/filters.pdf http://www.math.ntnu.no/emner/MA3002/2005v/ovinger/filtereng.pdf Dave L. Renfro
From: Victor Porton on 31 Mar 2010 17:39 On Apr 1, 12:08 am, "Dave L. Renfro" <renfr...(a)cmich.edu> wrote: > Victor Porton wrote (in part): > > > I want DIRECT proof that > > > (Quasi-)compactness (that is having every filter > > in X convergent) > > > is equivalent to > > > Every cover of the space X includes a finite cover of X. Oops, "Every open cover of the space X includes a finite cover of X." (the word "open" missed by me). > I don't know if a direct proof of what you want is in > either of the following, but they both seem worth looking at: > > http://www.efnet-math.org/~david/mathematics/filters.pdf > > http://www.math.ntnu.no/emner/MA3002/2005v/ovinger/filtereng.pdf Dave, there is the same problem with both articles you pointed that these prove closed-set versions of compactness (the same as Bourbaki) but I want a proof with open-sets (not closed sets). BTW, both articles you mentioned were already on my hard drive. You pointed me nothing new :-)
From: Dave L. Renfro on 31 Mar 2010 18:03 Victor Porton wrote (in part): > Dave, there is the same problem with both articles > you pointed that these prove closed-set versions of > compactness (the same as Bourbaki) but I want a proof > with open-sets (not closed sets). I thought this was the case, but I didn't have the time to look at them long enough to be sure (and right now I need to leave), but I did think the manuscripts would be useful to you (and I guess they are, since you said you already had copies of them). Dave L. Renfro
From: William Elliot on 1 Apr 2010 05:38
)n Wed, 31 Mar 2010, Victor Porton wrote: > On Apr 1, 12:08�am, "Dave L. Renfro" >> Victor Porton wrote (in part): >> >>> I want DIRECT proof that >> >>> (Quasi-)compactness (that is having every filter in X convergent) >> >>> is equivalent to >> >>> Every cover of the space X includes a finite cover of X. > > Oops, "Every open cover of the space X includes a finite cover of > X." (the word "open" missed by me). > >> I don't know if a direct proof of what you want is in >> either of the following, but they both seem worth looking at: >> > Dave, there is the same problem with both articles you pointed that > these prove closed-set versions of compactness (the same as Bourbaki) > but I want a proof with open-sets (not closed sets). > The equivalence of the open set and the closed set definitions of a compact space is straight forward basic logic and set theory using DeMorgan rules. > BTW, both articles you mentioned were already on my hard drive. You > pointed me nothing new :-) > The easiest direct proof is this two phase equivalence. |