Metric space

A metric space $(𝑀, 𝑑)$ is a set $𝑀$ equipped with a metric $𝑑 : 𝑀 \times 𝑀 \to [0, \infty)$ such that #m/def/anal

Symmetry: $𝑑 (𝑥, 𝑦) = 𝑑 (𝑦, 𝑥)$ for all $𝑥, 𝑦 \in 𝑀$
Triangle inequality: $𝑑 (𝑥, 𝑦) + 𝑑 (𝑦, 𝑧) \geq 𝑑 (𝑥, 𝑧)$ for all $𝑥, 𝑦, 𝑧 \in 𝑀$
Positive definite: $𝑑 (𝑥, 𝑦) = 0$ iff. $𝑥 = 𝑦$

It immediately follows that $𝑓 (𝑥, 𝑦) > 0$ iff. $𝑥 \neq 𝑦$ Metric spaces are the objects in the Category of metric spaces.

Examples

The quintessential example is the pythagorean distance function on euclidean space, which in one dimension is simply the difference

𝑑 : (𝑥 1, 𝑥 2) \mapsto | 𝑥 1 - 𝑥 2 |

A trivial example is the discrete metric, which yields the Discrete topology.

𝜌 (𝑥 1, 𝑥 2) = {1 i f 𝑥 \neq 𝑦 0 i f 𝑥 = 𝑦

Reverse triangle inequality
Metric spaces induce a Metric topology with the open balls as its Topological basis. Thus a metric space gives the most intuitive definition of open and closed sets, which is generalised by a topological space.

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