Math Is Fun Forum

bossk171

Member

Registered: 2007-07-16

Posts: 305

topology

I think I'm missing something really crucial, I can't get it at all:

Give and example of a set A on the real line with the usual topology, such that from A 14 distinct sets may be formed, using only the two operations of complementation and closure. More generally, show that in any space the maximum number of distinct sets which may be formed from a given set using only complementation and closure is 14.

What I tried:

Which gives 5.

The next set I tried was

when

and I'm not going to latex it up, but that only gave me 6. I have no idea how to show the general case.

Am I on the right track? am I way off? Is my notation ok? Also, how to I LaTeX the bar so that it covers more than one character?

Thanks a whole bunch.

EDIT: touched up LaTex

Last edited by bossk171 (2009-06-01 08:44:07)

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: topology

You forgot

The \bar isn't designed to cover more than one character, because it's more of a language command than a mathematical one. It's in the same family as \acute, \grave, etc.

\overline works fine though.

Regarding the actual question, I can't yet see a set that produces 14, but I think it's going to have to be more complicated than what you're doing.
The possible chains of operations are fairly restricting, since (A^c)^c = A and cl(cl(A)) = cl(A), and so it's pointless to put the same operation next to itself.
Not sure how to use that though.

My current best is

, which (I think) gives 8.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: topology

Let com(A) = complement of A and clo(A) = closure of A.  The key thing here is com(clo(com(A))) can produce a new set.

In your first example, com(clo(com(A))) = (a, b), which is not in your list.  Now clo(com(clo(com(A)))) = [a,b], which you already have.

For latex, the command \overline is almost always better than \bar.

You'll find that trying random sets won't get you very far, at least not very quickly.  What you want to do is look at things in a more general setting.

If we do com(clo(A)), we get every point which is not a limit point of A.  On the other hand, clo(com(A)) will give us every point which is not an interior point of A.

If we want to keep getting "new stuff", what has to happen?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: topology

If we do com(clo(A)), we get every point which is not a limit point of A.  On the other hand, clo(com(A)) will give us every point which is not an interior point of A.

If we want to keep getting "new stuff", what has to happen?

Points that are not limit points also need to be not interior points, right? I'm not sure if I know how to go about finding such a set...

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: topology

Ok, I think I got it, but if not, I'm close enough.  By combining your previous example, and mathsyperson's example (I started with just rationals instead of just irrationals), I got 10 different sets.

I think the thing here is to just be stupid and keep adding on different sets like (4, 5) and [6, 7].

Points that are not limit points also need to be not interior points, right?

Correct.

And just to inject my personal beliefs, this is not a very good topology question.  Interesting, sure, and perhaps a good math question in general, but it does not help you with topology much...

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: topology

And just to inject my personal beliefs, this is not a very good topology question.  Interesting, sure, and perhaps a good math question in general, but it does not help you with topology much...

Why is this? What would be a "good" topology question?

Also, what about the general case

More generally, show that in any space the maximum number of distinct sets which may be formed from a given set using only complementation and closure is 14.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: topology

For the general case, you want to do things... generally.  You have two operations, and from your example you can see exactly where your new sets are coming from.  The hope is that this is what always happens, that you will be able to prove that applying any more of the operations will give you something you already have.

Why is this? What would be a "good" topology question?

The standard topology of the real line should be used as a place where ideas come from (for example, the idea that a union of two opens sets is open).  On the other hand, problems in topology should show you the topological reasons for what things do hold on the real line, why things don't hold without certain assumptions (i.e. Hausdorff), or just investigations of weird spaces.

Again, this is just me and my personal opinion.  The more general question is better, and perhaps the example is necessary to find the general solution, but the problem is still rather tedious.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."