6

How can one know if a set is compact? From the definition, a set is compact if for any open cover, there exist a finite subcover. However, it is not possible to list out all the open covers to a set. So is there any way one can know if it is a compact set? In the case $R^n$ it is easy as one only needs to check the set is closed bounded but what about other metric space or other spaces?

Mathematics
  • 4,521
  • Do you have an example of a space that u want to show that it is compact – Amr Dec 07 '12 at 15:43
  • @Amr Is there any general way to do it? – Mathematics Dec 07 '12 at 15:44
  • 1
    I guess there isnt – Amr Dec 07 '12 at 15:45
  • Or a generL way might be checking the definition itself. But I don't know if there exists an equivalent definition that is somehow much easier to check – Amr Dec 07 '12 at 15:46
  • This relationship is 'iff'. The true power of defining it this way - and there has historically been debate about the best way of defining compact - is to use it if compact, then open cover... Using it the other way round can be tricky. I suggest you take amr up on what probably was an offer to show you how to use it to show compactness. This takes a bit time to sink in, but you'll soon find it natural. – gnometorule Dec 07 '12 at 15:47
  • @Amr What about a set of all continuous function in the interval $[0,1]$ with supremum norm metric? – Mathematics Dec 07 '12 at 15:50
  • 1
    @Mathematics: That is certainly not compact, since the sets ${f\mid \sup |f| < n}$ for $n\in\mathbb N$ cover the space but don't have a finite subcover. – hmakholm left over Monica Dec 07 '12 at 16:04
  • 1
    @Mathematics: See here for examples. – Mhenni Benghorbal Dec 07 '12 at 16:05

1 Answers1

8

An equivalent condition in the case of metric spaces is sequential compactness: Every sequence in the space has a convergent subsequence.

There exist a number of other characterizations, but they are usually not that easy to verify (such as being complete and totally bounded).

Often, one can use basic properties of compactness to show a given space is compact. For example, a closed subset of a compact set is compact and the forward image of a compact set under a continous function is compact.

There are many results for showing that some spaces are compact. These include the Arzelà–Ascoli theorem, the Banach–Alaoglu theorem, and a number of other results. These results are important precisely because it is in general often nontrivial to show that some space is compact.

Michael Greinecker
  • 32,841
  • 6
  • 80
  • 137