What is a Metric in Mathematics?

A metric is a way of measuring something. In mathematics, a metric is a function that assigns a distance between two points in a set. The distance is usually defined as the magnitude of the difference between the two points.

For example, consider the real numbers line, which is a set of all real numbers. We can define a metric on this set by using the absolute value of the difference between two real numbers. This means that the distance between two points x and y is given by |x - y|.

In more technical terms, a metric is a function d: X × X → R (where X is the set being measured and R is the set of real numbers) that satisfies the following three properties:

1. Non-negativity: d(x, y) ≥ 0 for all x, y in X and d(x, y) = 0 if and only if x = y.
2. Symmetry: d(x, y) = d(y, x) for all x, y in X.
3. Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z in X.

The metric space (X, d) is a set X together with a metric d that satisfies the above properties. The metric space can be used to define concepts such as distance, neighborhoods, and convergence.

In summary, a metric is a way of measuring distances between points in a set, and a metric space is a set together with a metric that defines the distances between points in the set.