Any idea that help in solving this problem is highly appreciated :)

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- Oct 16th 2011, 02:29 PMmia25Compactness over discrete metric space
Any idea that help in solving this problem is highly appreciated :)

- Oct 16th 2011, 02:49 PMPlatoRe: Compactness over discrete metric space
Question:

Is $\displaystyle \left\{ {\mathfrak{B}\left( {x;0.5} \right):x \in [0,1]} \right\}$ an open covering of $\displaystyle [0,1]~?$ - Oct 16th 2011, 03:04 PMmia25Re: 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 - Oct 16th 2011, 03:13 PMPlatoRe: Compactness over discrete metric space