Talk:Closed Bounded Subset of Real Numbers is Compact/Proof 1

Perhaps a "Closed [and bounded] interval in $\R$ is compact" page could be extracted from this, if it doesn't exist already? --Plammens (talk) 14:36, 14 April 2021 (UTC)

Oh nevermind, it *does* exist already: Closed Real Interval is Compact. I would suggest extracting the part of the proof that deals with this (i.e. everything after the first paragraph) into a new proof on that page, and cross-referencing that here. --Plammens (talk) 14:40, 14 April 2021 (UTC)


 * Good call. But there's a reason why Sutherland goes into the details of this proof from first principles. Can't remember exactly why now, probably to do with circularity. Bear in mind that there are various contexts in which compact is defined: in the real numbers, in a metric space, in the context of topology, and all these things have to be rigorously defined as being completely identical.


 * The latter is the Heine-Borel Theorem, which is a complicated beast, the basis of which is Closed Bounded Subset of Real Numbers is Compact.


 * Also note that Closed Real Interval is Compact is currently showing "Proof 1" and "Proof 2" which arises from a fundamental misunderstanding about the philosophy of . Proof 1 demonstrates the result in the context of metric spaces, Proof 2 demonstrates it in the context of normed vector spaces. It has not as far as I can tell, at that stage, been proved in the context of a topological space.


 * If it were really that simple and trivial to "just" invoke that result, then Sutherland would have done it. The fact that he takes great pains to go through the work that Heine and Borel themselves did to make sure that compact in the context of the real number line (and general multidimensional space) means the same thing as compact in the general topological space suggests that this is necessary -- or, at the very least, instructive.


 * Feel free to add a simple proof, though, based on your own ideas, but be aware that it may well either be incomplete (through reliance on results that have not actually been proven) or circular (through relying on this result itself). --prime mover (talk) 15:08, 14 April 2021 (UTC)