Question

Is the real line a metric space?

Answer 1

Answer on Question #71393, Math / Functional Analysis

Is the real line a metric space?

**Solution.** Yes, the real line is a metric space.

Recall that a set $X$ is a metric space if there is a function $d: X^2 \to \mathbf{P}$ such that for each $x, y, z \in X$:

(M1) $d(x,y)\geq 0$;

(M2) $d(x,y) = d(y,x)$;

(M3) $d(x,y)\leq d(x,z) + d(z,y)$;

(M4) $d(x,y) = 0 \Leftrightarrow x = y$.

Let $d: \mathbf{P}^2 \to \mathbf{P}$ be defined as:

$$d (x, y) = | x - y | \text{ for each } x, y \in \mathbf {P}.$$

(M1) $d(x,y)\geq 0$ because for each $x,y\in \mathbf{P}$;

(M2) $d(x,y) = |x - y| = |-(y - x)| = |y - x| = d(y,x)$;

(M3) $d(x,y) = |x - y| = |(x - z) + (z - y)| \leq |x - z| + |z - y| = d(x,z) + d(z,y)$;

(M4) $d(x,y) = 0 \Leftrightarrow |x - y| = 0 \Leftrightarrow x - y = 0 \Leftrightarrow x = y$.

Hence, the real line is a metric space.

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