Homework Help: HELP metric space problem

1. Feb 16, 2006

Mathman23

Hi

I have this here metric space problem which caused me some trouble:

$$S \subseteq \mathbb{R}^n$$ then the set

$$\{ \| x - y \| \ | y \in S \}$$ has the infimum

$$f(x) = \{ \| x - y \| \ | y \in S \}$$

where f is defined $$f: \mathbb{R}^n \rightarrow \mathbb{R}$$

I have two problems here which I'm unable to solve:

(a) show, if S is a closed set and $$x \notin S$$ then $$f(x) > 0$$ ????

(b) show, if S is a closed set, then $$S = \{ x \in \mathbb{R}^n | f(x) = 0\}$$ ???

I need to hand this in tomorrow, and I have been strugling this these two problems the last week, therefore I would very much appriciate if anybody could give me an idear on how to solve the two problems above.

God bless,

Best Regards,
Fred

Last edited: Feb 16, 2006
2. Feb 16, 2006

quasar987

$$f(x) = \{ \| x - y \| \ | y \in S \}$$ ??

That would mean to each x in R^n, f maps x to ||x-y|| for all y in S. So as soon as card(S)>1, f is not a function.

Also, what do you mean by "$$\{ \| x - y \| \ | y \in S \}$$ has the infimum f(x)"?

3. Feb 16, 2006

Mathman23

Sorry it should have said

$$f(x) = \mathrm{inf} \{ \| x - y \| \ | y \in S \}$$

Best Regards

Fred

p.s. My problems deals with the distance from $$\mathbb{R}^n$$ to a point in a subset S of $$\mathbb{R}^n$$.

Last edited: Feb 16, 2006
4. Feb 16, 2006

quasar987

a) wouldn't that exeedingly simply argument suffice:

we know that ||x-y|| = 0 iff x=y. But since x is not in S, x is not equal to y for any y in S. Hence, ||x-y||>0.

There's probably a problem with this argument as it doesn't even use the closedness of S...

5. Feb 16, 2006