# Compactness over discrete metric space

• October 16th 2011, 03:29 PM
mia25
Compactness over discrete metric space
Any idea that help in solving this problem is highly appreciated :)
• October 16th 2011, 03:49 PM
Plato
Re: Compactness over discrete metric space
Question:
Is $\left\{ {\mathfrak{B}\left( {x;0.5} \right):x \in [0,1]} \right\}$ an open covering of $[0,1]~?$
• October 16th 2011, 04:04 PM
mia25
Re: Compactness over discrete metric space

So you are saying that all the points are isolated point so we don't have a finite open cover of the set ?
Also, there is a theorem that states that every bounded and closed set in R^k is compact. so is the set of isolated point closed
• October 16th 2011, 04:13 PM
Plato
Re: Compactness over discrete metric space
Quote:

Originally Posted by mia25
So you are saying that all the points are isolated point so we don't have a finite open cover of the set ?

Correct.

Quote:

Originally Posted by mia25
Also, there is a theorem that states that every bounded and closed set in R^k is compact.

But that theorem is not for $R^k$ with a discrete metric.