# Thread: Compactness over discrete metric space

1. ## Compactness over discrete metric space

Any idea that help in solving this problem is highly appreciated

2. ## 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 $[0,1]~?$

3. ## Re: Compactness over discrete metric space

Thanks for your reply

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

4. ## Re: Compactness over discrete metric space

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.

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.