Metric spaces
Definition and Basics
A metric space is a set where a notion of distance (called a metric) between elements of the set is defined. The metric space concept in mathematics, particularly in the branches of topology, analysis, and algebra, serves as a fundamental building block in the study of abstract spaces. It is a crucial concept that underpins many mathematical disciplines and provides a framework for discussing continuity, convergence, and related concepts.
Metric
A metric on a set X is a function (called the distance function or simply distance)
- d : X × X → R (the set of real numbers)
where d(x,y) is the distance from x to y. For all x, y, z in X, this function is required to satisfy the following conditions:
1. d(x,y) ≥ 0 (non-negativity) 2. d(x,y) = 0 if and only if x = y (identity of indiscernibles. Note that condition 1 and 2 together produce positive definiteness) 3. d(x,y) = d(y,x) (symmetry) 4. d(x,z) ≤ d(x,y) + d(y,z) (subadditivity or triangle inequality).
Examples
There are numerous examples of metric spaces, each with unique properties and uses. Some of the most common metric spaces include:
1. The Euclidean space, which is probably the most familiar metric space. In this space, the distance between two points is given by the Pythagorean theorem.
2. The Discrete space, where the distance between two distinct points is always 1, and the distance from a point to itself is always 0.
3. The Manhattan distance, also known as taxicab distance, where the distance between two points is the sum of the absolute differences of their coordinates.
4. The Minkowski space, which is a vector space equipped with a non-degenerate, skew-symmetric bilinear form. This space is used in the theory of relativity.
Properties
Metric spaces have several important properties that make them a useful tool in mathematical analysis. These properties include:
1. Open sets: A set O in a metric space X is said to be open if for every point x in O there exists a real number r > 0 such that the open ball B(x, r) is completely contained in O.
2. Closed sets: A set C in a metric space X is said to be closed if it contains all its limit points.
3. Compactness: A metric space is said to be compact if every open cover has a finite subcover.
4. Connectedness: A metric space is said to be connected if it cannot be partitioned into two nonempty open sets.
Applications
Metric spaces are used in a wide range of mathematical and scientific fields. Some of the applications include:
1. In Computer Science, metric spaces are used in the study of algorithms, particularly those dealing with geometric problems.
2. In Physics, metric spaces are used in the theory of relativity, where spacetime is modeled as a four-dimensional metric space.
3. In Economics, metric spaces are used in the study of market equilibrium and optimization problems.