Compact Metric Spaces

1. Jun 10, 2009

Mathmos6

1. The problem statement, all variables and given/known data

Is the following statement true: for every compact metric space X there is a constant N S.T. every subcover of X by balls of radius one has a subcover with at most N balls?

2. Relevant equations

3. The attempt at a solution

I know you're meant to post your working but I really can't get started on this one! I can't even work out which way I should be proving, I have no clue whether this is true or false :( I have a feeling it's true but that's really got no actual mathematical basis sadly. I know the definitions of compactness - each open cover must have a finite subcover - and of balls, metric space etc, but I'm not sure how to apply it or how to approach the problem. Could someone please please get me started? Many many thanks,

Mathmos6

2. Jun 10, 2009

Dick

Maybe your feeling it's true is holding you back from trying to construct a counterexample. Think about a closed disk of radius 2. What you want to do is construct a series of open covers with the number of open sets needed to cover going to infinity.

Last edited: Jun 10, 2009