The dimension of a geometric figure is equal to one if the figure is a curve, two if the figure is a surface, and three if the figure is a solid. From the standpoint of analytic geometry, the dimension of a figure is equal to the number of coordinates needed to determine the position of a point on the figure. For example, the position of a point on a curve can be determined by a single coordinate, that of a point on a surface by two coordinates, and that of a point in three-dimensional space by three coordinates.
Until the mid-19th century, geometry dealt only with figures of three or less dimensions. With, however, the development of the concept of a multidimensional space, geometry began studying figures of any dimension. The simplest figures of dimension m are m-dimensional manifolds. An m-dimensional manifold in n-dimensional space is determined by n - m equations. For example, a curve, or one-dimensional manifold, is defined in three-dimensional space by 3 – 1 = 2 equations. The position of a point on an m-dimensional manifold is determined by what are called curvilinear coordinates. Thus, the position of a point on a sphere is determined by the point’s “geographic coordinates,” or latitude and longitude. The position of a point on a torus is defined in an analogous manner.
The above statements are valid only under certain restrictive assumptions. A truly general definition of the dimension of any closed bounded set lying in n-dimensional Euclidean space was given by P. S. Urysohn: the dimension of such a set is less than or equal to m if and only if the set admits of an ε-cover, for any ε > 0, by closed sets of multiplicity at most n + 1. This general definition of dimension can be extended in a natural way to extremely broad classes of topological spaces. In 1921, Urysohn constructed dimension theory, one of the most profound theories of modern topology. The further development of dimension theory has been due primarily to such Soviet mathematicians as P. S. Aleksandrov and L. S. Pontriagin.