Metric: Difference between revisions

Revision as of 00:22, 13 June 2026

In mathematics, a metric or distance function is a function that defines a rigorous concept of distance between each pair of elements in a given set. By generalizing the physical concept of distance, metrics allow mathematicians to define topological concepts such as open and closed sets, continuity, convergence, and limits in highly abstract spaces. A set equipped with a metric is called a metric space.

Definitions

Let $\text{[math]}$ be a given non-empty set. A metric on $\text{[math]}$ is a function

$\text{[math]}$

that satisfies the following four properties for all $\text{[math]}$ :

Non-negativity: $\text{[math]}$
Identity of indiscernibles: $\text{[math]}$
Symmetry: $\text{[math]}$
Triangle inequality: $\text{[math]}$ .

Properties

Property.

Non-negativity is implied by the other three properties.

Proof

Let $\text{[math]}$ . By triangle inequality,

$\text{[math]}$

By the identity of indisscernbles, $\text{[math]}$ . Thus

$\text{[math]}$

By symmetry, $\text{[math]}$ . Thus

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Reverse Triangle Inequality.

Suppose that $\text{[math]}$ is a metric space and $\text{[math]}$ . Then

$\text{[math]}$

{{Proof|proof=By triangle inequality,

$\text{[math]}$

By symmetry, re-arranging the inequality yields

$\text{[math]}$

Utilizing the triangle inequality again,

$\text{[math]}$

@@ Line 22: / Line 22: @@
    &0\le d(x,y)
 \end{align}</math>}}
+{{Property|title=Reverse Triangle Inequality|property=Suppose that <math>(X,d)</math> is a metric space and <math>x,y,z\in X</math>. Then <math display="block">|d(x,y)-d(x,z)|\le d(y,z).</math>}}
+{{Proof|proof=By triangle inequality,
+<math display="block">d(x,y)\le d(x,z)+d(z,y).</math>
+By symmetry, re-arranging the inequality yields
+<math display="block">d(x,y)-d(x,z)\le d(y,z).</math>
+Utilizing the triangle inequality again,
+<math display="block">d(x,z)\le d(x,y)+d(y,z).</math>