Space (mathematics)

Mathematics uses a wide assortment of spaces. Many of them are quite far from the ancient geometry.

Short history

In the ancient mathematics, "space" was a geometric abstraction of three-dimensional space observed in everyday life. Axiomatization of this space, started by Euclid, was finished in the 19th century. Non-equivalent axiomatic systems appeared in the same century: hyperbolic geometry (Nikolai Lobachevskii, János Bolyai, Carl Gauss) and 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 generally, 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).

Modern approach

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). We start with a basic class.

Space	Stipulates
Topological	Convergence, continuity. Open sets, closed sets.

Straight lines are defined in projective spaces. In addition, all questions applicable to topological spaces apply also to projective spaces, since each projective space (over the reals) "downgrades" to the corresponding topological space. Such relations between classes of spaces are shown below.

Space	Is richer than	Stipulates
Projective	Topological space.	Straight lines.
Affine	Projective space.	Parallel lines.
Linear	Affine space.	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 space. Metric space.	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 sub-covering.
	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 Andrei 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.
	σ-finite	The whole space is a countable union of sets of finite measure.
	finite	The whole space is of finite measure.
	Probability	The whole space is of measure 1.

These spaces are less geometric. In particular, the idea of dimension, applicable to topological spaces, therefore to all spaces listed in the previous tables, does not apply to measure spaces. Manifolds are much more geometric, but they are not called spaces. In fact, "spaces" are just mathematical structures (as defined by Nikolas Bourbaki) that often (but not always) are more geometric than other structures.