Heine-Borel Theorem: Bounded & Closed Set Covered by Finite Open Subsets

[pivoxa15]

The union of any collection of open sets WILL be open, where as the union of some collection of closed sets might be open, closed, both or neither.

The intersection of a finite number of open sets WILL be open, the intersection of an infinite number of closed sets WILL be closed.

The intersection of an infinite number of open sets might be open, closed, neither or both.

Let me tabulate all this to make everything clear

Union
Finite & open = open
Finite & closed = closed
Infinite & open = open
Infinite & closed = open, closed, neither or both

Intersection
Finite & open = open
Finite & closed = closed
Infinite & open = open, closed, neither or both
Infinite & closed = closedSo the interesting ones are An infinite union of closed sets and an intersection of infinitely many open sets.

 

[HallsofIvy]

Let me tabulate all this to make everything clear

Union
Finite & open = open
Finite & closed = closed
Infinite & open = open
Infinite & closed = open, closed, neither or both

Intersection
Finite & open = open
Finite & closed = closed
Infinite & open = open, closed, neither or both
Infinite & closed = closed


So the interesting ones are An infinite union of closed sets and an intersection of infinitely many open sets.

And such things are, in fact, called "Borel Sets".
http://mathworld.wolfram.com/BorelSet.html

 

[matt grime]

So the interesting ones are An infinite union of closed sets and an intersection of infinitely many open sets.

that would entirely depend on your definition of interesting, I think.

 

[pivoxa15]

What I forgot to ask was whether the other two infinite sets which are

Union
Infinite & open = open only?

Intersection
Infinite & closed = closed only?were open and closed respectively for all situations. In other words are they open only and closed only?

With a union of infinitely many open sets, could it be possible that it is closed or neither or both? For example, the union of all open sets in R is R which is clopen or both open and closed. So
Union
Infinite & open = open or both?

The intersection of finitely many closed sets seems to be closed for all situations hence closed only. But that is only if we do not include the empty set as a closed set. The empty set is both open and closed so we treat it as a clopen set only. I take it that the intersection of closed sets must have each set closed and closed only. Or is this not the case? Which would mean
Intersection
Infinite & closed = closed or both?I like to know all the possibilities.

 

[matt grime]

The union of infinitely many open sets must be open, you agreed that, so it is illogical to ask if the union of infinitely many open sets might be neither open nor closed. Also, you have seen by example here that the infinite union of open sets might also be closed.

The dual statements (changing open for closed and union for interesection) are also true.

 

[pivoxa15]

So there is a little bit of change. We have

Union
Finite & open = open
Finite & closed = closed
Infinite & open = open or closed&open
Infinite & closed = open, closed, neither or both

Intersection
Finite & open = open
Finite & closed = closed
Infinite & open = open, closed, neither or both
Infinite & closed = closed or closed&open

 

[loopgrav]

Why are any of you continuing with this idiotic thread? Don't you have anything else to do?

All of his questions have been answered repeatedly, and if nothing else I gave a detailed explanation along with several excellent texts that he could study on his own. In my opinion, this forum is to help others who make an honest effort to understand a subject, but just need a little guidance. This is the most inane thread I have ever seen. We are not here to hold someones hand, especially when they are clearly in way over their heads. In fact, I find it hard to believe he is serious.

 

[pivoxa15]

Why are any of you continuing with this idiotic thread? Don't you have anything else to do?

All of his questions have been answered repeatedly, and if nothing else I gave a detailed explanation along with several excellent texts that he could study on his own. In my opinion, this forum is to help others who make an honest effort to understand a subject, but just need a little guidance. This is the most inane thread I have ever seen. We are not here to hold someones hand, especially when they are clearly in way over their heads. In fact, I find it hard to believe he is serious.

All this stuff is very new to me and I am a slow learner. I have not done any set theory beyond year 8. I probably need a better grounding in set theory (espeically regarding infinite sets) before I can read any of those books properly. What do you think of the information in post 36? Is it complete?

 

[loopgrav]

I don't know what you mean by year 8. The only texts that I know of are college level and above. You may want to go to a nearby high school and talk to a math teacher if necessary to find a text. Alternatively, go to a nearby college bookstore and look at the course texts. If you wish, give me an email address and I will send you some chapters from a text that I wrote that was published some years ago but is now out of print. It's not on set theory, but it starts out with elementary set theory, and also discusses metric spaces in fairly complete detail as far as it goes.

 

[HallsofIvy]

All this stuff is very new to me and I am a slow learner. I have not done any set theory beyond year 8. I probably need a better grounding in set theory (espeically regarding infinite sets) before I can read any of those books properly. What do you think of the information in post 36? Is it complete?

I think what loopgrav was complaining about is that it not only is complete, it is "picky" What you really need to know is:
The union of any collection of open sets is open, the intersection of any finite collection of open sets is open
and that for closed sets it's just the opposite:
The union of any finite collection of closed sets is closed and the intersection of any collection of closed sets is closed.

There are too many interesting problems in topology to get bogged down in exactly what happens in other cases.

 

[loopgrav]

Pivoxa15:

What HallsofIvy says is exactly correct. Just apply the definition (in a metric space) of open to see that ANY collection of open sets is open, and a FINITE intersection of open sets is also open. THIS IS ALL YOU KNOW FOR SURE (plus the fact that by definition the empty set and the full space X are open). Everything else is a consequence of these two statements. You can't say anything else in general about unions or intersections -- each case must be handled individually.

To get to closed sets, use deMorgans laws (the complement of a union is the intersection of the complements) and the fact that a set is defined to be closed if its complement is open. For example, since the arbitrary (i.e. possibly infinite) union of open sets is open, its complement is closed. But the complement of the union of open sets is the intersection of closed sets, and hence the intersection of an arbitrary collection of closed sets is closed. Thus the only other thing you can say for sure about closed sets is that a finite union of closed sets is closed.

(At the risk of confusing you some more, if there exists a proper (i.e. neither empty nor X itself) subset of X that is both open and closed, then X is said to be disconnected. In other words, a set X is connected if the only subsets of X that are both open and closed are X and the empty set.)

I sent you an email with a link to download several chapters from my book. (In case you don't get it, it is http://download.yousendit.com/6F9375002C164119) It will be available for 7 days or 10 downloads, so I hope other people don't get them first. I would post them here but they are too large for this forum.

 

[mannyfold]

Keep in mind that Heine-Borel applies only to Euclidean space with the usual topology. It is a special, but important, case in topology.

 

[pivoxa15]

Pivoxa15:

What HallsofIvy says is exactly correct. Just apply the definition (in a metric space) of open to see that ANY collection of open sets is open, and a FINITE intersection of open sets is also open. THIS IS ALL YOU KNOW FOR SURE (plus the fact that by definition the empty set and the full space X are open). Everything else is a consequence of these two statements. You can't say anything else in general about unions or intersections -- each case must be handled individually.

To get to closed sets, use deMorgans laws (the complement of a union is the intersection of the complements) and the fact that a set is defined to be closed if its complement is open. For example, since the arbitrary (i.e. possibly infinite) union of open sets is open, its complement is closed. But the complement of the union of open sets is the intersection of closed sets, and hence the intersection of an arbitrary collection of closed sets is closed. Thus the only other thing you can say for sure about closed sets is that a finite union of closed sets is closed.

(At the risk of confusing you some more, if there exists a proper (i.e. neither empty nor X itself) subset of X that is both open and closed, then X is said to be disconnected. In other words, a set X is connected if the only subsets of X that are both open and closed are X and the empty set.)

I sent you an email with a link to download several chapters from my book. (In case you don't get it, it is http://download.yousendit.com/6F9375002C164119) It will be available for 7 days or 10 downloads, so I hope other people don't get them first. I would post them here but they are too large for this forum.

Thanks for the chapters. At my level I probably should think about things from first principles and worry about the details later.