Topology is a relatively young discipline of mathematics. Its roots reach back to the 19th century in which geometry underwent some major transformations. The classical interpretation of geometry at the beginning of the 19th century was that it described the real world. At the end of this century, however, other geometries were discovered (see historical development of geometry). Also, another important prerequisite for the development of modern topology was the progress made in set theory (see historical development of set theory). In particular, Georg Cantor (1845 - 1918), when studying ordinal numbers in the context of Fourier series, he discovered that some properties of subsets of $\mathbb R^n$ depended only on the underlying point sets. Later, Guido Ascoli (1887 - 1957), Vito Volterra (1860 - 1940) and Cesare Arzelà (1847–1912) replaced Cantor's point sets by function sets. They defined what we would today call a metric. The first attempts to define and use modern topological terms were done by René Maurice Fréchet (1878 - 1973) in the year 1906 in his thesis "Sur quelques points du calcul functionnel", in which he introduced the term metric space. He also attempts to create the term of a topological space, when trying to axiomatize convergent sequences of functions. While up to his time, different special cases of metric spaces were investigated, Fréchet realized that it can be very economical to use this abstract concept instead and get general and deeper results. For instance, instead of defining a convergent sequence $(x_n)_{n\in\mathbb N}$ "as usual", i.e. by saying that
... a sequence $(x_n)_{n\in\mathbb N}$ is convergent against $x$ if for every $\epsilon > 0$ there is a number $N\in\mathbb N$ such that $|x-x_n|<\epsilon$ for all n>N,$
Fréchet replaced the claim that the absolute value $|x-x_n|$ has to be less than a given $\epsilon > 0$ by the claim that in a metric space an abstractly defined distance has to fulfill this property.
Most of the concepts of the set-theoretic topology were first described in modern form in Felix Hausdorff's (1868 - 1942) book "Grundzüge der Mengenlehre" in its seventh chapter "Punktmengen in allgemeinen Räumen" ("Basics of the Set Theory", "Sets of Points in General Spaces"), published 1914 in Leipzig, but also again later by Fréchet, and by Frigyes Riesz (1880 - 1956). They all realized that not only the convergence of a sequence but also many other analytical concepts can be defined solely based on a single term of a neighborhood. In particular,
... a sequence $(x_n)_{n\in\mathbb N}$ is convergent against $x$ if for every neighborhood $U$ there is a number $N\in\mathbb N$ such that $x_n\in U$ for all n>N.$
The neighborhood itself was defined to fulfill certain properties and the subsets of the underlying sets were defined to be themselves neighborhoods. This was where birth of the topological space (1914). At one go it was possible to simplify the whole theory of convergence and to significantly extend its applicability.
Convergence was even more deeply understood after the concept of filters was later developed between 1935 and 1939. However, some analytical concepts, e.g. the concept of uniform continuity could not be covered by the new terminology. Only some decades later one succeeded to generalize these concepts too (uniform spaces).
The later development of topology was very rapid and was boosted by the magazine "Fundamenta Mathematicae" published first 1920 in Warsaw, in which, among others, Wacław Sierpiński (1882 - 1969) and Kazimierz Kuratowski (1896 - 1980) published some of their works. The 1920s developed also what was called combinatoric topology, or analysis situs, today better known as algebraic topology.
The so-called metrization problem:
Which are the purely topological properties a topological space must have to also have a metric?
was addressed by Pavel Urysohn (1898 - 1924) and Pavel Alexandrov (1896 - 1982), but solved only in 1951 by R. H. Bing (1914 - 1986), Juri Michailowitsch Smirnow (1921 - 2007) and Jun-iti Nagata (1925 - 2007).
In the undertaking Bourbaki, the view on topology as an underlying mathematical discipline was formulated. At the latest since then, topology has a set position among other mathematical disciplines.
