- #1
Pen_to_Paper
- 11
- 0
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!
"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!