Spaces have been seen as geometric abstractions of reality denoted in three dimensions and were described by the father of geometry, namely Euclid (325-270 BC) who created axioms around 300 BC. In 1637, René Descartes (1596-1650) exploited the insights of Euclid in his methods of coordinates. At this point in time, two major concepts were related with spaces, namely congruence and similarity.
Congruence can only be seen if two points are transformable into another using an isometry, which corresponds to a distance preserving transformation between two metric spaces (assumed to be bijective).
Similarity can be described as uniform scaling.
In 1795 Gaspard Monge (1746-1818) introduced the so called projective geometry, which resembles as third concept. This led to a development that was highly critical of Euclidean axioms. In 1816 Carl Friedrich Gauss (1777-1855), in 1829 Nikolai Lobachevsky (1792-1856) and in 1832 János Bolyai (1802-1860) worked on the concepts of non-Euclidean hyperbolic geometry. The existence of well-defined triangles bearing a sum of angles greater than 180° gave rise to problems since it was contradictory to Euclidean insights.
Eugenio Beltrami (1835-1900) and Felix Klein (1849-1925) resolved the issue in the years 1868 and 1871 respectively, which led to the abandonment of Euclidean geometry as “the absolute truth”. Additionally, it resulted in a change of view of axioms which henceforth were regarded as hypothesis instead.
The work of the group “Nicolas Bourbaki” as well as axioms from Hilbert, Tarshi and Birkhoff brought relevant reformations into the matter.