A metric space is a set of points equipped with a function called a metric, which measures the distance between any two points in the set. This metric must satisfy specific properties: it must be non-negative, equal to zero only when the points are the same, symmetric (the distance from point A to B is the same as from B to A), and satisfy the triangle inequality (the distance from A to C is less than or equal to the distance from A to B plus the distance from B to C). Metric spaces are fundamental in various areas of mathematics, including topology and analysis. They provide a framework for discussing concepts like convergence, continuity, and compactness. Common examples of metric spaces include the set of real numbers with the standard distance metric and the Euclidean space of points in geometry.