Space (mathematics)
A set with structure defining relationships among its elements.
In mathematics, a space is a set—sometimes called a universe—that comes with a structure describing how its elements relate to one another. A subspace is a subset of the original space that keeps the same mathematical structure. Modern mathematics uses many kinds of spaces, like Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, and probability spaces, but it never gives a single definition of "space" itself.
A space is made up of selected mathematical objects treated as points, along with chosen relationships between those points. The points can be very different things: numbers, functions on another space, or even subspaces of another space. What defines the space is the relationships. More precisely, isomorphic spaces are considered the same—an isomorphism is a one-to-one match between points that preserves the relationships. For instance, the relationships among points in a three-dimensional Euclidean space are fixed by Euclid's axioms, so all three-dimensional Euclidean spaces are considered identical.
Topological ideas like continuity have natural definitions in every Euclidean space. But topology doesn't tell straight lines apart from curved ones, so the link between Euclidean and topological spaces is "forgetful." These kinds of relationships are explored further in the "Types of spaces" section. It's not always clear whether a given mathematical object should be seen as a geometric "space" or an algebraic "structure." A general definition of "structure" proposed by Bourbaki covers all common types of spaces, gives a general definition of isomorphism, and allows properties to be transferred between isomorphic structures.
**History**
**Before the golden age of geometry**
In ancient Greek mathematics, "space" was a geometric abstraction of the three-dimensional reality of everyday life. Around 300 BC, Euclid set out axioms for the properties of space. He built all of mathematics on these geometric foundations, even defining numbers by comparing line segment lengths to a chosen reference segment. The coordinate method (analytic geometry) was adopted by René Descartes in 1637. At that time, geometric theorems were treated as absolute, objective truths known through intuition and reason, like objects of natural science, and axioms were seen as obvious implications of definitions.
Two equivalence relations between geometric figures
- field
- Mathematics
- known_for
- Foundational concept of mathematical space as a set with structure; distinction between Euclidean, projective, and non-Euclidean geometries; role of isomorphism and forgetful relations
Lore & Background
The concept of space in mathematics originated in ancient Greek geometry as an abstraction of three-dimensional reality. Euclid gave axioms for the properties of space around 300 BC, building all of mathematics on geometric foundations. The method of coordinates was adopted by René Descartes in 1637, treating geometric theorems as absolute objective truths. Two equivalence relations—congruence and similarity—were used for geometric figures, with a third, projective equivalence, introduced by Gaspard Monge in 1795.
Non-Euclidean hyperbolic geometry, introduced by Nikolai Lobachevsky in 1829 and János Bolyai in 1832 (and Carl Friedrich Gauss in 1816, unpublished), showed that the sum of a triangle's angles depends on the triangle and is always less than 180 degrees. Eugenio Beltrami in 1868 and Felix Klein in 1871 provided models for hyperbolic geometry. The period between 1795 and 1872 (Klein's Erlangen programme) is sometimes called the 'golden age of geometry' in historical contexts, though this term is not a standard designation from Bourbaki.
Reader's Guide
The concept of space is central to modern mathematics, though no single definition of 'space' itself exists. Instead, spaces are defined by their structure—the relationships among points—rather than the nature of the points. This allows points to represent numbers, functions, or subspaces of another space. Isomorphic spaces are considered identical, with an isomorphism being a one-to-one correspondence preserving relationships. The development of non-Euclidean geometries demonstrated that axioms are hypotheses, not absolute truths, and that mathematical objects are not given with their structure but are described by properties chosen as axioms. This led to the understanding that relations between objects are essential, while the nature of the objects is not. The shift from geometry to arithmetic as the foundation of mathematics, exemplified by Dedekind's work, redefined spaces as mathematical structures of convenience. Contemporary mathematicians follow Riemann's 1854 idea that any mathematical object parametrized by n real numbers may be treated as a point of an n-dimensional space, using classical geometry terminology nearly everywhere.
Did You Know?
- A space consists of selected mathematical objects treated as points and selected relationships between these points.
- Isomorphic spaces are considered identical; an isomorphism is a one-to-one correspondence between points that preserves the relationships.
- Non-Euclidean hyperbolic geometry, introduced by Lobachevsky in 1829 and Bolyai in 1832, showed that the sum of a triangle's angles is always less than 180 degrees.
- According to Bourbaki, the period between 1795 and 1872 is called 'the golden age of geometry'.
Frequently Asked Questions
What is Space (mathematics) in simple terms?
In mathematics, a space is essentially a collection of objects—treated as points—paired with a defined structure that tells you how those points relate to one another. Think of it as a universe where the 'laws' governing connections are baked into the definition itself.
What are the main types of mathematical spaces fans should know about?
The canon includes Euclidean spaces, linear (vector) spaces, topological spaces, Hilbert spaces, and probability spaces, among others. Notably, mathematics never pins down one universal definition of 'space'; each type carries its own structural rules.
How does a subspace work within a larger space?
A subspace is a subset of the original space that inherits and preserves the same mathematical structure. In other words, all the relational rules that apply in the parent space continue to hold inside the subspace without modification.
Why is the concept of Space (mathematics) considered foundational?
It provides the unifying framework that lets mathematicians talk about wildly different objects—numbers, functions, geometric figures—under one language of points and relationships. This framework underpins everything from Euclidean geometry to quantum mechanics and probability theory.
What role do isomorphism and forgetful relations play in the theory of spaces?
Isomorphism lets you show two different spaces are structurally identical, meaning their points and relationships correspond perfectly. Forgetful relations, by contrast, let you strip away extra structure to view a space as a 'simpler' one, revealing how richer spaces build on more basic ones.
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