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

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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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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 \}
Transcribed Image Text:**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 \}
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