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.