User:Boris Tsirelson/Sandbox1

       Space (mathematics)

In the ancient mathematics, "space" was a geometric abstraction of the three-dimensional space observed in the everyday life. Axiomatization of this space, started by Euclid, was finished in the 19 century. Non-equivalent axiomatic systems appeared in the same 19 century: the hyperbolic geometry (Nikolai Lobachevskii, Janos Bolyai, Carl Gauss) and the elliptic geometry (Georg Riemann). Thus, different three-dimensional spaces appeared: Euclidean, hyperbolic and elliptic. These are symmetric spaces; a symmetric space looks the same around every point.

Much more general, not necessarily symmetric spaces were introduced in 1854 by Riemann, to be used by Albert Einstein in 1916 as a foundation of his general theory of relativity. An Einstein space looks differently around different points, because its geometry is influenced by matter.

In 1872 the Erlangen program by Felix Klein proclaimed various kinds of geometry corresponding to various transformation groups. Thus, new kinds of symmetric spaces appeared: metric, affine, projective (and some others).

The distinction between Euclidean, hyperbolic and elliptic spaces is not similar to the distinction between metric, affine and projective spaces. In the latter case one wonders, which questions apply, in the former — which answers hold. For example, the question about the sum of the three angles of a triangle: is it equal to 180 degrees, or less, or more? In Euclidean space the answer is "equal", in hyperbolic space — "less"; in elliptic space — "more". However, this question does not apply to an affine or projective space, since the notion of angle is not defined in such spaces.

The classical Euclidean space is of course three-dimensional. However, the modern theory defines an $n$–dimensional Euclidean space as an affine space over an $n$–dimensional inner product space (over the reals); for $n=3$ it is equivalent to the classical theory.

Euclidean axioms leave no freedom, they determine uniquely all geometric properties of the space. More exactly: all three-dimensional Euclidean spaces are mutually isomorphic. In this sense we have "the" three-dimensional Euclidean space. Three-dimensional symmetric hyperbolic (or elliptic) spaces differ by a single parameter, the curvature. The definition of a Riemann space leaves a huge freedom, more than a finite number of numeric parameters. On the other hand, all affine (or projective) spaces are mutually isomorphic, provided that they are three-dimensional (or n-dimensional for a given n) and over the reals (or another given field of scalars).

Nowadays mathematics uses a wide assortment of spaces. Many of them are quite far from the ancient geometry. Here is a rough and incomplete classification according to the applicable questions (rather than answers). First, some basic classes.

Space	Stipulates
Projective	Straight lines.
Topological	Convergence, continuity. Open sets, closed sets.

Distances between points are defined in metric spaces. In addition, all questions applicable to topological spaces apply also to metric spaces, since each metric space "downgrades" to the corresponding topological space. Such relations between classes of spaces are shown below.

Space	Is richer than	Stipulates
Affine	Projective	Parallel lines
Linear	Affine	Origin. Vectors.
Linear topological	Linear space. Topological space.
Metric	Topological space.	Distances.
Normed	Linear topological space. Metric space.
Inner product	Normed space.	Angles.
Euclidean	Affine. Metric.	Angles.

A finer classification uses answers to some (applicable) questions.

Space	Special cases	Properties
Linear	three-dimensional	Basis of 3 vectors
	finite-dimensional	A finite basis
Metric	complete	All Cauchy sequences converge
Topological	compact	Every open covering has a finite subcovering
	connected	Only trivial open-and-closed sets
Normed	Banach	Complete
Inner product	Hilbert	Complete

Waiving distances and angles while retaining volumes (of geometric bodies) one moves toward measure theory and the corresponding spaces listed below. Besides the volume, a measure generalizes area, length, mass (or charge) distribution, and also probability distribution, according to Andrey Kolmogorov's approach to probability theory.

Space	Stipulates
Measurable	Measurable sets and functions.
Measure	Measures and integrals.

Measure space is richer than measurable space. Also, Euclidean space is richer than measure space.

Space	Special cases	Properties
Measurable	standard	Isomorphic to a Polish space with the Borel σ-algebra.
Measure	standard	Isomorphic mod 0 to a Polish space with a finite Borel measure.
Probability	The whole space is of measure 1.