- #1
Mathman23
- 254
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Hi
I have this here metric space problem which caused me some trouble:
[tex]S \subseteq \mathbb{R}^n[/tex] then the set
[tex]\{ \| x - y \| \ | y \in S \} [/tex] has the infimum [tex]f(x) = \{ \| x - y \| \ | y \in S \}[/tex]
where f is defined [tex]f: \mathbb{R}^n \rightarrow \mathbb{R}[/tex]
I have two problems here which I'm unable to solve:
(a) show, if S is a closed set and [tex]x \notin S [/tex] then [tex]f(x) > 0[/tex] ?
(b) show, if S is a closed set, then [tex]S = \{ x \in \mathbb{R}^n | f(x) = 0\}[/tex] ?
I need to hand this in tomorrow, and I have been strugling this these two problems the last week, therefore I would very much appreciate if anybody could give me an idear on how to solve the two problems above.
God bless,
Best Regards,
Fred
I have this here metric space problem which caused me some trouble:
[tex]S \subseteq \mathbb{R}^n[/tex] then the set
[tex]\{ \| x - y \| \ | y \in S \} [/tex] has the infimum [tex]f(x) = \{ \| x - y \| \ | y \in S \}[/tex]
where f is defined [tex]f: \mathbb{R}^n \rightarrow \mathbb{R}[/tex]
I have two problems here which I'm unable to solve:
(a) show, if S is a closed set and [tex]x \notin S [/tex] then [tex]f(x) > 0[/tex] ?
(b) show, if S is a closed set, then [tex]S = \{ x \in \mathbb{R}^n | f(x) = 0\}[/tex] ?
I need to hand this in tomorrow, and I have been strugling this these two problems the last week, therefore I would very much appreciate if anybody could give me an idear on how to solve the two problems above.
God bless,
Best Regards,
Fred
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