Exercise 3. [Closure in metric space] Let (X, d) be a metric space and Y be a nonempty subset of X. The distance of a point ze X from the subset Y is a function X → [0, +∞[ defined by d(x, y) = inf{d(x, y); y € Y}. 1. Verify that the distance function is well defined. 2. Prove that Y= {re X;d(x, y)=0}.
Exercise 3. [Closure in metric space] Let (X, d) be a metric space and Y be a nonempty subset of X. The distance of a point ze X from the subset Y is a function X → [0, +∞[ defined by d(x, y) = inf{d(x, y); y € Y}. 1. Verify that the distance function is well defined. 2. Prove that Y= {re X;d(x, y)=0}.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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topolgy exercice 3
![**Exercise 1: [Metric]**
Let \( p \) be a prime number, and \( d_p: \mathbb{Z} \times \mathbb{Z} \rightarrow [0, +\infty] \) be a function defined by
\[
d_p(x, y) = p^{-\max(m \in \mathbb{N}_0^+ \, \text{divides} \, (x-y))}.
\]
Prove that \( d_p \) is a metric on \( \mathbb{Z} \) and that \( d_p(x, y) \leq \max\{d_p(x, z), d_p(z, y)\} \), for every \(x, y, z \in \mathbb{Z}\).
**Exercise 2: [Closed in metric space]**
Let \((X, d)\) be a metric space and \(F \subset X\) a finite subset. Prove that \(F\) is closed in \(X\).
**Exercise 3: [Closure in metric space]**
Let \((X, d)\) be a metric space and \(Y\) be a nonempty subset of \(X\). The distance of a point \(x \in X\) from the subset \(Y\) is a function \(X \rightarrow [0, +\infty]\) defined by
\[
d(x, Y) = \inf\{d(x, y); y \in Y\}.
\]
1. Verify that the distance function is well defined.
2. Prove that \(\overline{Y} = \{x \in X; d(x, Y) = 0\}\).
**Exercise 4: [Separable space]**
Let \(X\) be a set of all real sequences \((x_n)_{n \in \mathbb{N}}\) converging to 0. Prove that the function
\[
d: X \times X \rightarrow [0, +\infty] \\
(x_n, y_n) \rightarrow d(x_n, y_n) = \sup_{n \in \mathbb{N}} |x_n - y_n|
\]
is a metric on \(X\). Show that the metric space \((X, d)\) is separable.
**Exercise](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc44ee4c-f0ee-41b9-a7bf-13469d6ba446%2F0b6c081d-9ac1-4b77-a384-bd44549b1e45%2F1ps37tn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 1: [Metric]**
Let \( p \) be a prime number, and \( d_p: \mathbb{Z} \times \mathbb{Z} \rightarrow [0, +\infty] \) be a function defined by
\[
d_p(x, y) = p^{-\max(m \in \mathbb{N}_0^+ \, \text{divides} \, (x-y))}.
\]
Prove that \( d_p \) is a metric on \( \mathbb{Z} \) and that \( d_p(x, y) \leq \max\{d_p(x, z), d_p(z, y)\} \), for every \(x, y, z \in \mathbb{Z}\).
**Exercise 2: [Closed in metric space]**
Let \((X, d)\) be a metric space and \(F \subset X\) a finite subset. Prove that \(F\) is closed in \(X\).
**Exercise 3: [Closure in metric space]**
Let \((X, d)\) be a metric space and \(Y\) be a nonempty subset of \(X\). The distance of a point \(x \in X\) from the subset \(Y\) is a function \(X \rightarrow [0, +\infty]\) defined by
\[
d(x, Y) = \inf\{d(x, y); y \in Y\}.
\]
1. Verify that the distance function is well defined.
2. Prove that \(\overline{Y} = \{x \in X; d(x, Y) = 0\}\).
**Exercise 4: [Separable space]**
Let \(X\) be a set of all real sequences \((x_n)_{n \in \mathbb{N}}\) converging to 0. Prove that the function
\[
d: X \times X \rightarrow [0, +\infty] \\
(x_n, y_n) \rightarrow d(x_n, y_n) = \sup_{n \in \mathbb{N}} |x_n - y_n|
\]
is a metric on \(X\). Show that the metric space \((X, d)\) is separable.
**Exercise
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