Math Is Fun Forum

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Interval

Prove that every closed set in R^1 is the intersection of a countable collection of open sets.

I know that

but I don't exactly know how to do it.

Last edited by Stanley_Marsh (2007-05-21 18:56:52)

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Numbers are the essence of the Universe

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: Interval

Maybe

Can I use induction , assuming S can be the intersection of n open sets , then consider n+1 ?

Oh , I think I have a better idea , since every open sets can be a intersection of a collection of countable open set. right?

Last edited by Stanley_Marsh (2007-05-21 19:41:01)

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Numbers are the essence of the Universe

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Interval

A finite intersection of open sets will only result in an open set. (This is part of the definition of a topological space.) In

the only sets that both open and closed are Ø and

(as

is a connected space). So if your closed S is not Ø or

it must be the intersection of an infinite collection of open sets.

I can’t see how induction is going to help here.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Interval

Well, I’m as much stumped by the question as you – so here is the outline of a proof I found by Googling.

Last edited by JaneFairfax (2007-05-22 00:05:06)

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Interval

First, [a2,a1] is included in or a subset of the two.

Second, any number out of [a1,a2] is doom to fail to fall into  both two of the original intervals.

Done.

Last edited by George,Y (2007-05-22 01:43:44)

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X'(y-Xβ)=0

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Interval

Stan, one thing you must stop doing is thinking about everything in intervals.  Intervals are not the only closed sets in R.  In fact, they are extremely boring.  Instead, always go to the definitions.

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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..."

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: Interval

Thanks , I am studying the proof   , I used to do maths like Geometry , Calculus .  I havent really gotten used to  Abstract maths.

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Numbers are the essence of the Universe

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Interval

A logic book may help you.

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X'(y-Xβ)=0

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Interval

Intervals are not that humble, Ricky. Probability theory depends on them.

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X'(y-Xβ)=0

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Interval

Probably so.  But we are talking about analysis here George.  When it comes to analysis, intervals are boring.

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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..."

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: Interval

Logic book? there is this kinda stuff? thanks

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Numbers are the essence of the Universe