(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex](X,d)[/tex] be a metric space, and let [tex]A,B \subset X[/tex] be disjoint closed subsets.

1. Construct a continuous function [tex]f : X \to [0,1][/tex] such that [tex]A \subseteq f^{-1}({0})[/tex] and [tex]B \subseteq f^{-1}({1})[/tex]. Hint: use the functions below.

2. Prove that there are disjoint sets [tex]U,V \subset X[/tex] such that [tex]A \subset U[/tex] and [tex]B \subset V[/tex].

2. Relevant equations

[tex]f(x) := d(x,A) = \inf \{ d(x,y) | y \in A \}[/tex] is uniformly continuous on [tex]X[/tex].

3. The attempt at a solution

1. I see that for points in [tex]A[/tex] the hint function will always return [tex]0[/tex]. So the preimage of [tex]\{ 0 \}[/tex] is set [tex]A[/tex]. But how can I make the maximum distance from any point to [tex]A[/tex] to [tex]1[/tex] and keep the function continuous? And also how should I make this function return [tex]1[/tex] for all points in [tex]B[/tex]?

2. I don't understand this (second) part of the question. Is this related to part one? Otherwise it seems obviously true.

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# Homework Help: Construct a continuous function in metric space

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