
1.3: Metric spaces


The notion of metric space provides a rigorous way to say: “we can mea- sure distances between points”. That is, instead of (i) on Section 1.1, we can say “Euclidean plane is a metric space”.

Definition

Let $$\mathcal{X}$$ be a nonempty set and $$d$$ be a function which returns a real number $$d(A, B)$$ for any pair $$A, B \in \mathcal{X}$$. Then $$d$$ is called metric on $$\mathcal{X}$$ if for any $$A, B, C \in \mathcal{X}$$, the following conditions are satisfied:

(a) Positiveness:

$d(A, B) \ge 0.$

(b) $$A = B$$ if and only if

$d(A, B) = 0.$

(c) Symmetry:

$d(A, B) = d(B, A)$

(d) Triangle inequality:

$d(A, C) \le d(A, B) + d(B, C).$

A metric space is a set with a metric on it. More formally, a metric space is a pair $$(\mathcal{X}, d)$$ where $$\mathcal{X}$$ is a set and $$d$$ is a metric on $$\mathcal{X}$$.

The elements of $$\mathcal{X}$$ are called points of the metric space. Given two points $$A, B \in \mathcal{X}$$, the value $$d(A, B)$$ is called distance from $$A$$ to $$B$$.

Example $$\PageIndex{1}$$ Discrete metric

Let $$\mathcal{X}$$ be an arbitrary set. For any $$A, B \in \mathcal{X}$$, set $$d(A, B) = 0$$ if $$A = B$$ and $$d(A, B) = 1$$ otherwise. The metric $$d$$ is called discrete metric on $$\mathcal{X}$$.

Example $$\PageIndex{2}$$ Real line

Set of all real numbers ($$\mathbb{R}$$) with metric $$d$$ defined by

$d(A, B) := |A - B|.$

Exercise $$\PageIndex{1}$$

Show that $$d(A, B) = |A - B|^2$$ is not a metric on $$\mathbb{R}$$.

Metrics on the plane. Suppose that $$\mathbb{R}^2$$ denotes the set of all pairs $$(x, y)$$ of real numbers. Assume $$A = (x_A, y_A)$$ and $$B = (x_B, y_B)$$. Consider the following metrics on $$\mathbb{R}^2$$:

• Euclidean metric, denoted by $$d_2$$, and defined as
$d_2(A, B) = \sqrt{(x_A - x_B)^2 + (y_A - y_B)^2}.$
• Manhattan metric, denoted by $$d_1$$ and defined as
$d_1(A, B) = |x_A - x_B| + |y_A - y_B|.$
• Maximum metric, denoted by $$d_{\infty}$$ and defined as
$d_{\infty}(A, B) = \max \{|x_A - x_B|, |y_A - y_B|\}.$
Hint

Check the triangle inequality for $$A = 0$$, $$B = 1$$ and $$C = 2$$.

Exercise $$\PageIndex{2}$$

Prove that the following functions are metrics on $$\mathbb{R}^2$$:

(a) $$d_1$$;
(b) $$d_2$$;
(c) $$d_{\infty}$$.

Only the triangle inequality requires a proof — the rest of conditions in Definition 1.1 are evident. Let $$A = x_A, y_A)$$, $$B = (x_B, y_B)$$, and $$C = (x_C, y_C)$$. Set

$$x_1 = x_B - x_A$$,                     $$y_1 = y_B - y_A$$,
$$x_2 = x_C - x_B$$,                     $$y_2 = y_C - y_B$$.

(a). The inequality

$d_1(A, C) \le d_1(A, B) + d_1(B, C)$

can be written as

$|x_1 + x_2| + |y_1 + y_2| \le |x_1| + |y_1| + |x_2| + |y_2|.$

The latter follows since $$|x_1 + x_2| \le |x_1| + |x_2|$$ and $$|y_1 + y_2| \le |y_1| + |y_2|$$.

(b). The inequality

$d_2(A, C) \le d_2 (A, B) + d_2(B, C)$

can be written as

$\sqrt{(x_1 + x_2)^2 + (y_1 + y_2)^2} \le \sqrt{x_1^2 + y_1^2} + \sqrt{x_2^2 + y_2^2}.$

Take the square of the left and the right hand sides, simplify, take the square again and simplify again. You should get the following inequality

$0 \le (x_1 \cdot y_2 - x_2 \cdot y_1)^2,$

which is equivalent to 1.3.9 and evidently true.

(c). The inequality

$d_{\infty} (A, C) \le d_{\infty} (A, B) + d_{\infty} (B, C)$

can be written as

$\max \{|x_1 + x_2|, |y_1 + y_2|\} \le \max \{|x_1|, |y_1|\} + \max \{|x_2|, |y_2|\}.$

Without loss of generality, we may assume that

$\max \{|x_1 + x_2|, |y_1 + y_2|\} = |x_1 + x_2|.$

Further,

$|x_1 + x_2| \le |x_1| + |x_2| \le \max \{|x_1|, |y_1|\} + \max \{|x_2|, |y_2|\}.$

Hence 1.3.13 follows.