Re: separation of compact and closed set

On Sat, 12 Nov 2005 berthuffman@xxxxxxxxx wrote:

> I know this is easy, but it is catching me for some reason.
>
> Let K be compact and F be closed in some metric space. I want to show
> that K and F are disjoint iff the distance between the two is greater
> than zero.
>
Here's another similar proof.
Assuming disjoint F,K.
As F is closed, x in F iff d(x,F) = 0

Thus for all x in K, d(x,F) > 0.
As K compact, some k in K with d(k,F) = inf{ d(x,F) | x in K }
= inf{ d(x,y) | x in K, y in F } = distance between K and F
Finally as d(k,F) > 0, distance between K and F > 0.

The converse is easy.

.