Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![Let \( A \) be a non-empty subset of a metric space \((X, d)\) and \( x \) an element of \( X \). Define the distance from \( x \) to \( A \) as
\[
d(x, A) = \inf\{d(x, a) : a \in A\}.
\]
(i) Prove that the function \( f_A : X \to \mathbb{R} \), defined as \( f_A(x) = d(x, A) \), satisfies
\[
|f_A(x) - f_A(y)| \leq d(x, y) \quad \forall x, y \in X,
\]
and that \( f_A \) is continuous on \( X \).
(ii) Prove that \( \overline{A} = \{x : x \in X \text{ and } f_A(x) = 0\} \).
(iii) Suppose \( A \) and \( B \) are nonempty disjoint closed subsets of \( X \). Use the function \( g := f_A - f_B \) to prove that there exist disjoint open sets \( U \) and \( V \) with \( A \subseteq U \) and \( B \subseteq V \).](https://content.bartleby.com/qna-images/question/e15ed467-90ec-4e60-afef-3d3f6119f74d/fe79c758-1032-4c81-91bf-3bfa4da61c35/9rsah8zb_processed.png)
Transcribed Image Text:Let \( A \) be a non-empty subset of a metric space \((X, d)\) and \( x \) an element of \( X \). Define the distance from \( x \) to \( A \) as
\[
d(x, A) = \inf\{d(x, a) : a \in A\}.
\]
(i) Prove that the function \( f_A : X \to \mathbb{R} \), defined as \( f_A(x) = d(x, A) \), satisfies
\[
|f_A(x) - f_A(y)| \leq d(x, y) \quad \forall x, y \in X,
\]
and that \( f_A \) is continuous on \( X \).
(ii) Prove that \( \overline{A} = \{x : x \in X \text{ and } f_A(x) = 0\} \).
(iii) Suppose \( A \) and \( B \) are nonempty disjoint closed subsets of \( X \). Use the function \( g := f_A - f_B \) to prove that there exist disjoint open sets \( U \) and \( V \) with \( A \subseteq U \) and \( B \subseteq V \).
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