Space (mathematics): Difference between revisions

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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 <math>n</math>–dimensional Euclidean space as an affine space over an <math>n</math>–dimensional inner product space (over the reals); for <math>n=3</math> 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==
== Modern approach==
Nowadays mathematics uses a wide assortment of spaces. Many of them
Nowadays mathematics uses a wide assortment of spaces. Many of them

Revision as of 14:10, 7 July 2009

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.

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