Could you explain how to show 9.8 in detail? **Theorem 9.8.** A metric space is Hausdorff, regular, and normal.**Definition.** A **metric** on a set \( M \) is a function \( d : M \times M \rightarrow \mathbb{R}_+ \) (where \( \mathbb{R}_+ \) is the non-negative real numbers) such that for all \( a, b, c \in M \), these properties hold:1. \( d(a, b) \geq 0 \), with \( d(a, b) = 0 \) if and only if \( a = b \);2. \( d(a, b) = d(b, a) \);3. \( d(a, c) \leq d(a, b) + d(b, c) \).These three properties are often summarized by saying that a metric is **positive definite, symmetric,** and satisfies the **triangle inequality**.A **metric space** \( (M, d) \) is a set \( M \) with a metric \( d \).**Example.** The function \( d(x, y) = |x - y| \) is a metric on \( \mathbb{R} \). This measure of distance is the **standard metric** on \( \mathbb{R} \).**Example.** On any set \( M \), we can define the **discrete metric** as follows: for any \( a, b \in M \), \( d(a, b) = 1 \) if \( a \neq b \) and \( d(a, a) = 0 \). This metric basically tells us whether two points are the same or different.**Example.** Here’s a strange metric on \( \mathbb{Q} \): for reduced fractions, let \( d\left(\frac{a}{b}, \frac{m}{n}\right) = \max(|a - m|, |b - n|) \). Which rationals are “close” to one another under this metric?**Theorem 9.3.** Let \( d \) be a metric on the set \( X \). Then the collection of all open balls\[ \mathcal{B} = \{B(p, \epsilon) = \{y \in X | d(p, y) < \epsilon \}

The objective is to show that every metric is hausdorff, regular and normal. Let x≠y be points of a…

Answered: Theorem 9.8. A metric space is…

Theorem 9.8. A metric space is Hausdorff, regular, and normal.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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