Talk:Continuum (order theory)
how can a continuum be compact? There is no finite subcover of the Real line, the classical example of the continuum. This would make more sense if it was locally compact, no? jordan 13:19, 28 July 2006 (UTC)
It seems to me, real line is actually also compact connected metric space. The first meaning would be actually only an instance of the second one. AsiBakshish 16:49, 13 April 2009 (UTC)
The real line is most certainly not compact; that has a specialized meaning that every open cover has a finite subcover; since the cover of the real line by intervals of the form (n, n+2) has no subcover, it is not compact.
What's happened is that "continuum" has two meanings, and unfortunately both have survived in mathematics. The one pertaining to ordered sets need not be compact; the topological definition includes things like the closed unit interval, closed unit disc, pseudoarc and closed topologist's sine curve, but not the real line.--Syd Henderson (talk) 00:53, 8 July 2009 (UTC) _____________________________________________________________________________________________________________________ May I ask for more obviously outlining the promised two distinct meanings of continuum in mathematics? Also, I cannot find a reference explaining the word shah. How to pronounce this word?
According to Peirce who was influenced by Leibniz, a continuum is something every part of which has parts. I would like to call such continuum the ideal one, corresponding to the ideal and original notion of infinity being not considered as a quantity but as a fictitious quality that cannot get enlarged and also not get exhausted.