- #31
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Well, it follows immediately that A is countable, not? Since both B and B' are countable basis?
What is the countable basis problem in topology?
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SUMMARY
The countable basis problem in topology asserts that if a topological space X has a countable basis {Bn}, then every basis C for X must also contain a countable basis for X. The discussion elaborates on constructing a countable subcollection C' from an open cover C, demonstrating that C' covers X and satisfies the necessary conditions to be a basis. The participants clarify the implications of intersections of basis elements and the necessity of ensuring that the resulting collection generates the correct topology, particularly in relation to the Euclidean topology.
PREREQUISITES- Understanding of basic topology concepts, including bases and open covers.
- Familiarity with second-countable spaces and their properties.
- Knowledge of intersection properties of sets in topology.
- Proficiency in using lemmas related to bases in topology, specifically Lemma 13.2 from Munkres' Topology.
- Study the concept of second-countable spaces in detail.
- Learn about the properties of bases in topology, focusing on Lemma 13.2 from Munkres' Topology.
- Explore examples of bases for different topologies, particularly the Euclidean topology.
- Investigate the implications of countable unions and intersections in the context of topological spaces.
Mathematicians, students of topology, and anyone interested in understanding the foundational concepts of topological spaces and their bases.
- #32
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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?
- #33
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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...
- #34
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radou said:
Yes, I think you've got it!
- #35
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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?
- #36
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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.
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.
- #37
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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...
- #38
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radou said: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...
- #39
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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.
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.
- #40
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Of course, all these are closely related, so it's probably all a matter of personal choice and taste.
- #41
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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..?
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..?
- #42
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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.
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.
- #43
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micromass said:
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.
- #44
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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?
If we have a countable collection of sets, is the number of all possible unions of these sets countable?
- #45
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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.
- #46
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Yes, you'e correct. It isn't true 
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