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December 14th, 2008, 08:41 PM   #1
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The Real Line & Countable Complement Topology- not Compact?

Hi everyone,

How would you prove that the Real Line equipped with the Countable Complement Topology is NOT Compact? Obviously the method should be to construct an open cover without ?nite subcovers, but I can't quite how one would go about doing this?

Many thanks in advance.
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December 22nd, 2008, 04:30 PM   #2
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Re: The Real Line & Countable Complement Topology- not Compact?

Right its not compact. Here is a proof:

I will denote the rationals by Q.

Consider the set {q}U(Q)^c where q is in Q [so a point in Q union every point not in Q]

Taking this sets complement gives Q\{q} which is clearly countable.

Then we can do this for each point in q to get an open cover of R.

But if were to suppose R with this topology were compact then there would be only finitely many open sets and thus we would only get finitely points in Q which is impossible since we must cover R and therefore Q as well.

Q.E.D.

There was nothing special about Q in this proof really. We could have used any [b]inifintely[b] countable set such as the set of all algebraic numbers in place of Q in the proof.
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