Metric space

Definition

A Metric space is a set where a notion of distance (called a metric) between elements of the set is defined. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally: the distance from point A to point B is zero if and only if A and B are the same point, the distance from point A to point B is the same as the distance from B to A, and the distance from point A to point C via B is at most the distance from A to B plus the distance from B to C.

An illustration of points in a metric space, demonstrating the concept of distance between points.

Properties

Metric spaces are characterized by four properties. If M is a set and d: M × M → R is a function, then (M, d) is a metric space if and only if for any x, y, z ∈ M, the following four conditions are satisfied:

1. d(x, y) ≥ 0 (non-negativity) 2. d(x, y) = 0 if and only if x = y (identity of indiscernibles) 3. d(x, y) = d(y, x) (symmetry) 4. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

These properties make metric spaces a generalization of the real line, where the metric is given by the absolute difference, and of Euclidean space, where the metric is given by the Euclidean distance.

Examples

There are many examples of metric spaces. Some of these include:

- The set of real numbers, with the metric d(x, y) = |x - y|. - The Euclidean plane, with the Euclidean distance metric. - The set of continuous functions on a closed interval [a, b], with the metric d(f, g) = max{|f(x) - g(x)| : x in [a, b]}.

Topology of Metric Spaces

The metric topology on a metric space M is the collection of all sets that can be expressed as the union of open balls in M. An open ball in M is a set of all points in M that are less than a certain distance from a certain point of M. The metric topology is the coarsest topology that makes the metric a continuous function on the product space M × M.

Continuity

In a metric space, a function f: M → N between two metric spaces is continuous if for every point x in M and every positive real number ε, there is a positive real number δ such that the distance from x to any point in M within δ implies that the distance from f(x) to f(y) is less than ε.

Compactness

In metric spaces, compactness and sequential compactness are equivalent properties. A metric space M is compact if every sequence of points in M has a subsequence that converges to a point in M. This property is a generalization of the Heine-Borel theorem in Euclidean space.

Convergence

In a metric space, a sequence of points is said to converge if the distance between successive terms can be made arbitrarily small by going sufficiently far out in the sequence. The limit of a sequence is the point that the sequence converges to.