Math and science::Topology

# Metric space

#### Metric

A metric $$d$$ on a set $$X$$ is a function [ $$d : \text{what} \to \text{ to what?}$$] with the following three properties:

What is this called?
$$d(x, y) = 0 \iff x = y, \text{ for all } x, y \in X$$
Triangle inequality
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Symmetry
$$d(x, y) = d(y, x), \text{ for all } x, y \in X$$

A metric space is a set together with a metric on it, or more formally, a pair $$(X, d)$$ where $$X$$ is a set and $$d$$ is a metric on $$X$$.