Understanding Compact Sets: Exploring the Definition and Examples

[felani]

Hello physicsforum -

I recently began self-studying real analysis on a whim and I have run into some trouble understanding the idea of compactness. I have no accessible living sources near me and I have yet to find a source that has explained the matter clearly, so I shall turn here for help.

From my primitive understanding of classifying a set as compact, I understand that every open cover of said set has to have a finite sub-cover that still covers that set. So, in order to declare a set as non-compact, it must be shown that there exists some open cover that has no finite sub-cover. Take, for instance, the closed interval [0, 1]: from what I've read, this is a compact set. A possible open cover for this set would simply be the ball centered at x = 1/2 with a radius greater than 1/2. This is an open cover that has no finite sub-cover - it is a single set so it has no sub-covers! How can [0, 1] be compact if it has an open cover that has no finite sub-cover?

I realize that the previous example might be special since it consists of a singular set as the open cover, and maybe that is what tripped me up; but, here is another example: consider the finite set {1, 2, 3}. From what I've read, all finite sets are compact, so therefore this set is compact. But take the open cover {B₁, B₂, B₃}, where B_i is an open ball with radius 0.1 centered at x = i. If you consider any sub-cover of this open cover, you must remove one of the sets in the open cover. Once you do, your finite sub-cover does not cover the entire set! Again, how can any finite set be compact if it can be shown that there exists an open cover that does not have a finite sub-cover?

I'm either missing something or drawing some wildly incorrect conclusions - if anyone could offer any insight to clear up my muddled thinking I would greatly appreciate it.

Thanks!

 

[tornado28]

You are being tripped up by the difference between a subset and a proper subset. If S is a set then U is a subset of S if each element of U is also in S. Thus every set is a subset of itself, and the empty set is a subset of every set. A proper subset is a subset that is not equal to the original set. The definition of compact allows non-proper subsets.

If you're really interested in compactness you should find an introductory book on Topology. Munkres is pretty good.

 

[felani]

Ah, that makes sense. I had forgotten about that.

Thank you!

And yes I may have to find some other topology sources - I'm working my way through Rudin's Principles of Mathematical Analysis right now (accompanied by whatever else I can find online).