working with closed covers and finite sets

• Oct 16th 2012, 06:07 PM
bskcase98
working with closed covers and finite sets
I am not sure how to proceed with this proof.

Say that a set K in a metric space is 'Tompac' if every closed cover of K has a finite subcover. show that such a K must be a finite set.
(hint: any singleton set in a metric space is ...)

I know that any singleton set in a metric space is closed. my thinking on this is that a closed set contains the set and its limit points so if the set has a closed subcover say B[p, epsilon] then the set must be a subset of the cover but I am not sure how to show the set must be finite.
• Oct 16th 2012, 06:53 PM
johnsomeone
Re: working with closed covers and finite sets
Quote:

Originally Posted by bskcase98
I know that any singleton set in a metric space is closed.

With that observation, the problem is now actually easier than you suspect. It's now just a matter of seeing the right "bad" closed cover of K that will lead you to the conclusion.

With that observation, think of a really bad, meaning really big, closed cover of K. Try to make the cover as big as possible - meaning containing as many closed sets as possible. Do so in a way that has as little overlap as possible.

OK - now that cover you just produced has a *finite* subcover. Some finite number of those closed sets in your cover are sufficient to still cover K. What does that tell you about K?
• Oct 16th 2012, 07:17 PM
bskcase98
Re: working with closed covers and finite sets
so let each x in M in the metric (M,d) have a closed cover G then the union of all the covers will be big enough to cover K then any finite subcover that covers K must imply that K is finite.
• Oct 16th 2012, 07:35 PM
johnsomeone
Re: working with closed covers and finite sets
Quote:

Originally Posted by bskcase98
so let each x in M in the metric (M,d) have a closed cover G then the union of all the covers will be big enough to cover K then any finite subcover that covers K must imply that K is finite.

I didn't understand that. Are you describing a particular cover? If so, can you describe your particular cover exactly?