Wednesday, December 22, 2021

Space in mathematics

Space is a set wit
h some added structure.

While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of “space” itself.

A space consists of selected mathematical objects that are treated as points, and selected relationships between these points.

The nature of the points can vary widely: for example, the points can be elements of a set, functions on another space, or subspaces of another space. It is the relationships that define the nature of the space.

More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional Euclidean space are uniquely determined by Euclid’s axioms, and all three-dimensional Euclidean spaces are considered identical.

Topological notions such as continuity have natural definitions in every Euclidean space. However, topology does not distinguish straight lines from curved lines, and the relation between Euclidean and topological spaces is thus “forgetful”.

We classify spaces on three levels. Given that each mathematical theory describes its objects by some of their properties.

We classify spaces on three levels. Given that each mathematical theory describes its objects by some of their properties, the first question to ask is: which properties? This leads to the first (upper) classification level. On the second level, one takes into account answers to especially important questions (among the questions that make sense according to the first level). On the third level of classification, one takes into account answers to all possible questions.

For example, the upper-level classification distinguishes between Euclidean and projective spaces, since the distance between two points is defined in Euclidean spaces but undefined in projective spaces. Another example. The question “what is the sum of the three angles of a triangle” makes sense in a Euclidean space but not in a projective space. In a non-Euclidean space the question makes sense but is answered differently, which is not an upper-level distinction.

The second-level classification distinguishes, for example, between Euclidean and non-Euclidean spaces; between finite-dimensional and infinite-dimensional spaces; between compact and non-compact spaces, etc.

The third-level classification distinguishes, for example, between spaces of different dimension, but does not distinguish between a plane of a three-dimensional Euclidean space, treated as a two-dimensional Euclidean space, and the set of all pairs of real numbers, also treated as a two-dimensional Euclidean space.

Likewise it does not distinguish between different Euclidean models of the same non-Euclidean space.

It is not always clear whether a given mathematical object should be considered as a geometric “space”, or an algebraic “structure”. A general definition of “structure”, proposed by Bourbaki, embraces all common types of spaces, provides a general definition of isomorphism, and justifies the transfer of properties between isomorphic structures.

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