Space (mathematics)
In mathematics, a space is a set (sometimes known as a universe) with a definition (structure) of relationships among the elements of the set. 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.[1][a]
For other uses, see Space (disambiguation).
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 represent numbers, 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,[b] and all three-dimensional Euclidean spaces are considered identical.
Topological notions such as continuity have natural definitions for every Euclidean space. However, topology does not distinguish straight lines from curved lines, and the relation between Euclidean and topological spaces is thus "forgetful". Relations of this kind are treated in more detail in the "Types of spaces" section.
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,[2] embraces all common types of spaces, provides a general definition of isomorphism, and justifies the transfer of properties between isomorphic structures.
Taxonomy of spaces[edit]
Three taxonomic ranks[edit]
While each type of space has its own definition, the general idea of "space" evades formalization. Some structures are called spaces, other are not, without a formal criterion. Moreover, there is no consensus on the general idea of "structure".
According to Pudlák,[7] "Mathematics [...] cannot be explained completely by a single concept such as the mathematical structure. Nevertheless, Bourbaki's structuralist approach is the best that we have."
We will return to Bourbaki's structuralist approach in the last section "Spaces and structures", while we now outline a possible classification of spaces (and structures) in the spirit of Bourbaki.
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.
Also, the distinction between a Euclidean plane and a Euclidean 3-dimensional space is not an upper-level distinction; the question "what is the dimension" makes sense in both cases.
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.
In Bourbaki's terms,[2] the second-level classification is the classification by "species". Unlike biological taxonomy, a space may belong to several species.
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.
More formally, the third level classifies spaces up to isomorphism. An isomorphism between two spaces is defined as a one-to-one correspondence between the points of the first space and the points of the second space, that preserves all relations stipulated according to the first level. Mutually isomorphic spaces are thought of as copies of a single space. If one of them belongs to a given species then they all do.
The notion of isomorphism sheds light on the upper-level classification. Given a one-to-one correspondence between two spaces of the same upper-level class, one may ask whether it is an isomorphism or not. This question makes no sense for two spaces of different classes.
An isomorphism to itself is called an automorphism. Automorphisms of a Euclidean space are shifts, rotations, reflections and compositions of these. Euclidean space is homogeneous in the sense that every point can be transformed into every other point by some automorphism.
Euclidean axioms[b] 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. In Bourbaki's terms, the corresponding theory is univalent. In contrast, topological spaces are generally non-isomorphic; their theory is multivalent. A similar idea occurs in mathematical logic: a theory is called categorical if all its models of the same cardinality are mutually isomorphic. According to Bourbaki,[8] the study of multivalent theories is the most striking feature which distinguishes modern mathematics from classical mathematics.
Relations between species of spaces[edit]
Topological notions (continuity, convergence, open sets, closed sets etc.) are defined naturally in every Euclidean space. In other words, every Euclidean space is also a topological space. Every isomorphism between two Euclidean spaces is also an isomorphism between the corresponding topological spaces (called "homeomorphism"), but the converse is wrong: a homeomorphism may distort distances. In Bourbaki's terms,[2] "topological space" is an underlying structure of the "Euclidean space" structure. Similar ideas occur in category theory: the category of Euclidean spaces is a concrete category over the category of topological spaces; the forgetful (or "stripping") functor maps the former category to the latter category.
A three-dimensional Euclidean space is a special case of a Euclidean space. In Bourbaki's terms,[2] the species of three-dimensional Euclidean space is richer than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space.
Such relations between species of spaces may be expressed diagrammatically as shown in Fig. 3. An arrow from A to B means that every A-space is also a B-space, or may be treated as a B-space, or provides a B-space, etc. Treating A and B as classes of spaces one may interpret the arrow as a transition from A to B. (In Bourbaki's terms,[9] "procedure of deduction" of a B-space from a A-space. Not quite a function unless the classes A,B are sets; this nuance does not invalidate the following.) The two arrows on Fig. 3 are not invertible, but for different reasons.
The transition from "Euclidean" to "topological" is forgetful. Topology distinguishes continuous from discontinuous, but does not distinguish rectilinear from curvilinear. Intuition tells us that the Euclidean structure cannot be restored from the topology. A proof uses an automorphism of the topological space (that is, self-homeomorphism) that is not an automorphism of the Euclidean space (that is, not a composition of shifts, rotations and reflections). Such transformation turns the given Euclidean structure into a (isomorphic but) different Euclidean structure; both Euclidean structures correspond to a single topological structure.
In contrast, the transition from "3-dim Euclidean" to "Euclidean" is not forgetful; a Euclidean space need not be 3-dimensional, but if it happens to be 3-dimensional, it is full-fledged, no structure is lost. In other words, the latter transition is injective (one-to-one), while the former transition is not injective (many-to-one). We denote injective transitions by an arrow with a barbed tail, "↣" rather than "→".
Both transitions are not surjective, that is, not every B-space results from some A-space. First, a 3-dim Euclidean space is a special (not general) case of a Euclidean space. Second, a topology of a Euclidean space is a special case of topology (for instance, it must be non-compact, and connected, etc). We denote surjective transitions by a two-headed arrow, "↠" rather than "→". See for example Fig. 4; there, the arrow from "real linear topological" to "real linear" is two-headed, since every real linear space admits some (at least one) topology compatible with its linear structure.
Such topology is non-unique in general, but unique when the real linear space is finite-dimensional. For these spaces the transition is both injective and surjective, that is, bijective; see the arrow from "finite-dim real linear topological" to "finite-dim real linear" on Fig. 4. The inverse transition exists (and could be shown by a second, backward arrow). The two species of structures are thus equivalent. In practice, one makes no distinction between equivalent species of structures.[10] Equivalent structures may be treated as a single structure, as shown by a large box on Fig. 4.
The transitions denoted by the arrows obey isomorphisms. That is, two isomorphic A-spaces lead to two isomorphic B-spaces.
The diagram on Fig. 4 is commutative. That is, all directed paths in the diagram with the same start and endpoints lead to the same result. Other diagrams below are also commutative, except for dashed arrows on Fig. 9. The arrow from "topological" to "measurable" is dashed for the reason explained there: "In order to turn a topological space into a measurable space one endows it with a σ-algebra. The σ-algebra of Borel sets is the most popular, but not the only choice." A solid arrow denotes a prevalent, so-called "canonical" transition that suggests itself naturally and is widely used, often implicitly, by default. For example, speaking about a continuous function on a Euclidean space, one need not specify its topology explicitly. In fact, alternative topologies exist and are used sometimes, for example, the fine topology; but these are always specified explicitly, since they are much less notable that the prevalent topology. A dashed arrow indicates that several transitions are in use and no one is quite prevalent.
This article was submitted to WikiJournal of Science for external academic peer review in 2017 (reviewer reports). The updated content was reintegrated into the Wikipedia page under a CC-BY-SA-3.0 license (2018). The version of record as reviewed is:
Boris Tsirelson; et al. (1 June 2018). "Spaces in mathematics" (PDF). WikiJournal of Science. 1 (1): 2. doi:10.15347/WJS/2018.002. ISSN 2470-6345. Wikidata Q55120290.
