# Metric A **metric** defines what distance is in a space and thereby defines the geometry of that space. 1. Metric: In mathematics, a metric on a set is a function that defines a distance between any two elements of the set. A metric space is a set equipped with such a distance function. 2. Properties of a Metric: A function d is a metric if it satisfies the following properties for all points x, y, z in the space: * **Non-negativity**: $d(x, y) \geq 0$ * **Identity of Indiscernibles**: $d(x, y) = 0$ if and only if $x = y$ * **Symmetry**: $d(x, y) = d(y, x)$ * **Triangle Inequality**: $d(x, z) \leq d(x, y) + d(y, z)$ ### Examples of metrics * Euclidean Space: In three-dimensional Euclidean space, the metric is given by $ds^2 = dx^2 + dy^2 + dz^2$ , which corresponds to the familiar distance formula. * Minkowski Space: In special relativity, the Minkowski metric $ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$ defines the geometry of spacetime, where distances and intervals are invariant under Lorentz transformations. > Thus, by defining how distances are measured, a metric determines the fundamental geometric properties of a space, shaping its structure and the relationships between points within it. This is really the big idea here - we start with a set of points. We can just think of these points as objects with properties. We *naturally* think about the euclidean distance between them. But we could absolutely pick some other metric to capture the distance between these points! That is our *choice*. --- Date: 20240612 Links to: Tags: References: * []()