Space (mathematics)

In mathematics, a space is a set (sometimes called a universe) with some added structure. In mathematics, a space is a set (sometimes called a universe) with 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'. Relations of this kind are treated in more detail in the Section 'Types of spaces'. 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. In ancient Greek mathematics, 'space' was a geometric abstraction of the three-dimensional reality observed in everyday life. About 300 BC, Euclid gave axioms for the properties of space. Euclid built all of mathematics on these geometric foundations, going so far as to define numbers by comparing the lengths of line segments to the length of a chosen reference segment. The method of coordinates (analytic geometry) was adopted by René Descartes in 1637. At that time, geometric theorems were treated as absolute objective truths knowable through intuition and reason, similar to objects of natural science;:11 and axioms were treated as obvious implications of definitions.:15 Two equivalence relations between geometric figures were used: congruence and similarity. Translations, rotations and reflections transform a figure into congruent figures; homotheties — into similar figures. For example, all circles are mutually similar, but ellipses are not similar to circles. A third equivalence relation, introduced by Gaspard Monge in 1795, occurs in projective geometry: not only ellipses, but also parabolas and hyperbolas, turn into circles under appropriate projective transformations; they all are projectively equivalent figures. The relation between the two geometries, Euclidean and projective,:133 shows that mathematical objects are not given to us with their structure.:21 Rather, each mathematical theory describes its objects by some of their properties, precisely those that are put as axioms at the foundations of the theory.:20