in mathematics, a logically conceivable form or structure that is used as a setting in which other forms and various constructions are realized. For example, in elementary geometry the plane or space is the setting in which various figures are constructed. In most spaces, we introduce relations whose formal properties are similar to those of ordinary spatial relations, such as distance between points or congruence of figures. Consequently, such spaces may be said to represent logically conceivable spacelike forms.
Historically, the first mathematical space was three-dimensional Euclidean space, which is an approximate abstract image of physical space; it has remained a very important space in mathematics. The general concept of space took shape in mathematics as a result of the gradual, increasingly broad generalization and modification of the concepts of the geometry of Euclidean space. The first spaces differing from three-dimensional Euclidean space were introduced in the first half of the 19th century. These spaces were Lobachevskian space and n-dimensional Euclidean space. The general concept of mathematical space was advanced in 1854 by B. Riemann. The process of generalizing, refining and concretely defining the concept followed various directions; for example, such concepts as vector space, Hilbert space, Riemannian space, function space, and topological space were developed.
In contemporary mathematics a space is defined as a set of objects, which are called the points of the space. These objects may be. for example, geometric figures, functions, or the states of a physical system. When we consider a set of objects as a space, we deal not with the individual properties of the objects but with only those properties of the set that are determined by relations that we wish to take into account or that we introduce by definition. These relations between points and various configurations, or sets of points, determine the geometry of the space. When the geometry is constructed axiomatically, the basic properties of these relations are expressed in the corresponding axioms.
Three examples of spaces are metric spaces, spaces of events, and phase spaces. In a metric space, the distance between points is defined. Thus, the functions f(x) continuous on an interval [a, b] form a metric space—whose points are the functions f (x)— when the distance between f1(x) and f2(x) is defined as the maximum of the absolute value of the difference between the two functions:
r = max ǀf1(x) –f2(x)ǀ
The concept of space of events plays an important role in the geometric interpretation of the theory of relativity. Every event is characterized by its position—the coordinates x, y, and z— and the time t of its occurrence. The set of all possible events is thus a four-dimensional space, in which an event or point is defined by the four coordinates x, y, z, and t.
Phase spaces are studied in theoretical physics and mechanics. The phase space of a physical system is the set of all the possible states of the system. The states are the points of the space.
The spaces in these examples are of significance in the actual universe since the set of possible states of a physical system or the set of events with space and time coordinates has real existence. Consequently, we are dealing with forms of reality that, although not spatial in the ordinary sense, are spacelike in structure. The question of which mathematical space reflects most accurately the general properties of physical space is answered experimentally. Thus, it has been established that in describing physical space Euclidean geometry is not always sufficiently accurate, and Riemannian geometry is used in the present-day theory of physical space (seeRELATIVITY, THEORY OF). The concept of space in mathematics is also discussed in the articles GEOMETRY, MATHEMATICS, and MULTIDIMENSIONAL SPACE.
A. D. ALEKSANDROV