Could you help me better understand covers, open covers, subcovers ?

[Pen_to_Paper]

First off, I'm using Rudin in my Analysis class, and although I have the written definition of a cover, I don't quite grasp it:

"An open cover of a set E in a metric space X is a collection of {G sub-alpha} (G indexed by alpha's) of open subsets of X such that E is contained in the union of the collection of open subsets of G indexed the alpha's"
(3rd edition of Principles/Analysis; Definition 2.31)

and then the extension (Heine-Borel) that a subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover

I'm really trying to intuitively grasp this defintion and I went through some of the older posts discussing subcovers, but that didn't really illuminate the meaning for me.First, in the definition of open cover, the set E may or may not be open, is this correct?

so can be E = (1,2) or E=[1,2] and then an open cover of E may be this: let H = (1/2, 3/2) U (4/3, 7/3) ?

Thus, the set H forms an open cover of E ? Granted, this is a very simple example but I want to be able to grasp it at layman's terms so that I don't assume the wrong things for a large set of G-alpha's (large indexing set)Ok, now, onto subcover. Is a subcover open or closed? or does the term cover always imply that said cover is open?

Does the term subcover simply mean a subset of covers? I realize that in terms of the definition of compactness, a compact subset K is contained in the union of finitely many indices, alpha-1 to alpha-n, so now, this might be a trivial question but I'm just trying to think about it in terms of the real number line so here goes; if there are infinitely many numbers in (0,1) aren't there infinitely many numbers in [0,1]? (Yes, I know one is a closed interval and the other open) if so, wouldn't there still be infinitely many non-empty unions of subsets in [0,1]?

I know I am wrong because of the very definition of compactness (the definition in terms of finite subcover) but I'm trying to grasp it intuitively. For example; I like the idea of explaining compactness like this: in a closed set, [a,b] a<b, if you start at one point and take infinitely many steps in one direction, you will get (some arbitrary distance) closer to another point; by extension, it's easy to see why the real line isn't compact. But obviously I'm in Analysis because I also want to be able to extend the definitions to other proofs, not just simple descriptions.

I just try to start with simple examples just to make sure I'm clear on the meanings of these definitions, and I'm not just memorizing their notations.Any help with simple, numerical examples will be much appreciated - thank you for your help in advance!

 

[micromass]

First, in the definition of open cover, the set E may or may not be open, is this correct?

Correct.

so can be E = (1,2) or E=[1,2] and then an open cover of E may be this: let H = (1/2, 3/2) U (4/3, 7/3) ?

Thus, the set H forms an open cover of E ?

Correct. But allow me to pedantic here. In the definition you supplied, you have only defined a cover for a metric space. Here you are taking a cover of a subset of the metric space \mathbb{R}.

Of course, you can view E as a metric space itself. But then the sets in your cover need to be open sets IN E! Thus an open cover of [1,2] will consist of the sets [1,3/2) and (4/3,2].

Ok, now, onto subcover. Is a subcover open or closed? or does the term cover always imply that said cover is open?

Does the term subcover simply mean a subset of covers?

Yes, a subcover is just a subset of the cover such that this subset still covers the entire space.

For example, {(0,1),(1/2,2),(1/2,3/2)} will be an open cover of (0,2). A subcover would be
{(0,1),(1/2,2)}. A subcover would NOT be {(0,1),(1/2,3/2)}.

That said, all the elements of an open cover are open. Thus a subcover of an open cover is again an open cover. If the cover is not necessairily open, then the subcover doesn't need to be open to (but it can be).

I realize that in terms of the definition of compactness, a compact subset K is contained in the union of finitely many indices, alpha-1 to alpha-n, so now, this might be a trivial question but I'm just trying to think about it in terms of the real number line so here goes; if there are infinitely many numbers in (0,1) aren't there infinitely many numbers in [0,1]? (Yes, I know one is a closed interval and the other open) if so, wouldn't there still be infinitely many non-empty unions of subsets in [0,1]?

Yes, [0,1] may have infinitely many open subsets. But the point is, that among these subsets, you will find finitely many which will still cover [0,1].

Take (0,1), an open cover of this is {(1/2,1),(1/4,1),(1/8,1),...}. But this does not have an open subcover (thus we cannot find finitely many elements in this cover which still cover (0,1) ).

The example you should start of with is the difference between \{0\}\cup\{1/n~\vert~n&gt;0\} and \{1/n~\vert~n&gt;0\}.
We can find an open cover of the second set which does not have an open subcover (try to find one!). But we cannot do this for the first set (even though it has more elements!). Indeed, take an open cover {G_i}_i of the first set. Then there must exist an i such that G_i contains 0. But since 0 is the limit of the sequence, there must be an N, such that 1/n\in G_i for all i>N. So all but finitely many indices remain uncovered. But we can easily cover these points by finitely many sets. Thus we obtain an finite subcover.

 

[Pen_to_Paper]

Thank you! I found this part in particular very, very helpful:

"Yes, a subcover is just a subset of the cover such that this subset still covers the entire space.

For example, {(0,1),(1/2,2),(1/2,3/2)} will be an open cover of (0,2). A subcover would be
{(0,1),(1/2,2)}. A subcover would NOT be {(0,1),(1/2,3/2)}.

That said, all the elements of an open cover are open. Thus a subcover of an open cover is again an open cover. If the cover is not necessairily open, then the subcover doesn't need to be open to (but it can be)."I had seen the {0}U{1/n | n>0} and {1/n | n>0} example before (online) but I feel like I can really appreciate the difference now; "So all but finitely many indices remain uncovered." It never occurred to me that all but the finitely many indices remain uncovered (perhaps I was reading it wrong?)

Nonetheless, I really appreciate your examples! Thank you, thank you!