What is the countable basis problem in topology?

[radou]

Well, it follows immediately that A is countable, not? Since both B and B' are countable basis?

 

[radou]

Hence, the collestion C is countable. Now, if we apply our lemma here, let U be any open set and x some element in U, then ma post #28 implies that there is a C from the collection A such that x is in C and C is in U, hence C is a countable basis?

 

[micromass]

Yes, countability is easy. The fun part is to prove that its a basis. But I don't think its to hard. The hardest part is finding what the countable basis is supposed to be...

 

[micromass]

Hence, the collestion C is countable. Now, if we apply our lemma here, let U be any open set and x some element in U, then ma post #28 implies that there is a C from the collection A such that x is in C and C is in U, hence C is a countable basis?

Yes, I think you've got it!

 

[radou]

This seems too simple...The only thing we need to apply is the lemma (see my post #32) and we know that the countable collection C is a basis?

 

[radou]

Ah, OK, thanks a lot!

The real solutions to such problems are actually always quite simple, but require a certain amount of creativity. The proof I tried is more "definition-based", and such proofs ofteh lead to nothing.

 

[micromass]

The hardest part was finding what the countable basis is. Proving that its a basis is indeed not so difficult. Sadly this is typical for topology, once you know what the things are supposed to be, it isn't hard to prove that they are...

 

[micromass]

Ah, OK, thanks a lot!

The real solutions to such problems are actually always quite simple, but require a certain amount of creativity. The proof I tried is more "definition-based", and such proofs ofteh lead to nothing.

Well the problem here is that Munkres didn't define basis in a very good way. I always define basis as lemma 13.2, since that is the form one will always use when discussing a basis...

 

[radou]

Yes, it's interesting how definitions differ from author to author.

In a set of lecture notes on metric spaces and topology I went through earlier, the definition of a basis for a topology T is that it's a subfamily B of T such that every member of T equals a union of the members of B.

In Munkres for example, this is stated as a separate lemma.

Back in this set of lecture notes, the definition from Munkres is actually stated as a theorem.

 

[radou]

Of course, all these are closely related, so it's probably all a matter of personal choice and taste.

 

[radou]

By the way, could we define the family A as follows: (?)

Let A be the family of all such basis elements B for which there is some basis element C contained in them. Now take the family of all the basis elements C contained in some B. This family is countable.

Now, if U is any open set, and x in U, there exists some basis element B containing x. Further on, there exists some basis element C containing x and contained in B, hence this C belongs to the earlier defined countable family, which is by this argument a basis for out topology.

Frankly, I don't see a conceptual difference between this "proof" and the last one..?

 

[micromass]

I suppose that argument would work out to...

I'm still wondering if you can't prove the problem directly from the definition of a basis. I would deem it possible, but it would be more difficult.

 

[radou]

I suppose that argument would work out to...

I'm still wondering if you can't prove the problem directly from the definition of a basis. I would deem it possible, but it would be more difficult.

OK.

I wrote that down, I didn't give up on the other proof attempt, if I figure something out, I'll post it here.

 

[radou]

I may have an idea, but before I consider it, I have the following question (it may seem a bit stupid, but nevertheless):

If we have a countable collection of sets, is the number of all possible unions of these sets countable?

 

[radou]

Actually, I think the answer is no; I just found that the power set of the positive integers is uncountable, and my question is equivalent to asking if the power set of the positive integers is countable.

 

[micromass]

Yes, you'e correct. It isn't true