why is compactness important

[remaths]

Hi everyone,

I am asking very basic question. But I think there is some important concept I am missing.

why is compactness important or why should one look out for compact sets?

I understand that they are bounded and closed and functions on them have good properties wrt to continuity and maxima or minima.


thanks
remaths

[HallsofIvy]

I've said this before but it bears repeating: compactness is "the next best thing to finite"! Consider all of the properties of a finite set: It is closed; it is bounded; in the case of a metric space, there exist two points in the set having maximum distance between them; in the case of a finite set of real numbers it contains a smallest and a largest number. All of those are true for a compact set. In fact, in each case you can prove them by taking an open cover consisting of open sets about each point in the set and using compactness to reduce to a finite cover- and so to a finite set of points.

[remaths]

Thank you for your reply

[rochfor1]

compactness is "the next best thing to finite"

This is one of the most amazing one-line summaries of a mathematical concept I have ever seen. I was about to say this about 100 times less eloquently.

[ice109]

halls beat me to it but i was going to parrot my analysis teacher and say exactly that