Space (mathematics)

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.