Exercise 2. [Closed in metric space] Let (X, d) be a metric space and FX be a finite subset. Prove that F is closed in X.
Exercise 2. [Closed in metric space] Let (X, d) be a metric space and FX be a finite subset. Prove that F is closed in X.
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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topology exercice 2
![**Exercise 1. [Metric]** Let \( p \) be a prime number, and define \( d_p: \mathbb{Z} \times \mathbb{Z} \rightarrow [0, +\infty] \) by
\[
d_p(x, y) = p^{-\max(m \in \mathbb{N}_0 : p^m \text{ divides } (x-y))}
\]
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 \( F \subseteq X \) be 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 \( X \). Define the distance of a point \( x \in X \) from the subset \( Y \) as a function \( X \to [0,+\infty] \) 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%2Fb03c98a1-d28d-4d38-b5b0-2a6edea8d209%2F5bykbhf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise 1. [Metric]** Let \( p \) be a prime number, and define \( d_p: \mathbb{Z} \times \mathbb{Z} \rightarrow [0, +\infty] \) by
\[
d_p(x, y) = p^{-\max(m \in \mathbb{N}_0 : p^m \text{ divides } (x-y))}
\]
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 \( F \subseteq X \) be 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 \( X \). Define the distance of a point \( x \in X \) from the subset \( Y \) as a function \( X \to [0,+\infty] \) 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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