Editing Topological space (section)

== History ==
Around 1735, [[Leonhard Euler]] discovered the [[Planar graph#Euler's formula|formula]] <math>V - E + F = 2</math> relating the number of vertices (V), edges (E) and faces (F) of a [[Convex polytope|convex polyhedron]], and hence of a [[planar graph]]. The study and generalization of this formula, specifically by [[Augustin-Louis Cauchy| Cauchy]] (1789–1857) and [[Simon Antoine Jean L'Huilier| L'Huilier]] (1750–1840), [[Euler's Gem | boosted the study]] of topology. In 1827, [[Carl Friedrich Gauss]] published ''General investigations of curved surfaces'', which in section 3 defines the curved surface in a similar manner to the modern topological understanding: "A curved surface is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitesimal distance from A are deflected infinitesimally from one and the same plane passing through A."{{sfn|Gauss|1827}}{{psi|date=June 2024}}

Yet, "until [[Bernhard Riemann| Riemann]]'s work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered".{{sfn|Gallier|Xu|2013}} "[[August Ferdinand Möbius| Möbius]] and [[Camille Jordan| Jordan]] seem to be the first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are [[homeomorphism | homeomorphic]] or not."{{sfn|Gallier|Xu|2013}}

The subject is clearly defined by [[Felix Klein]] in his "[[Erlangen Program]]" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced by [[Johann Benedict Listing]] in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by [[Henri Poincaré]]. His first article on this topic appeared in 1894.<ref>J. Stillwell, Mathematics and its history</ref> In the 1930s, [[James Waddell Alexander II]] and [[Hassler Whitney]] first expressed the idea that a surface is a topological space that is [[Topological manifold |locally like a Euclidean plane]].

Topological spaces were first defined by [[Felix Hausdorff]] in 1914 in his seminal "Principles of Set Theory". [[Metric spaces]] had been defined earlier in 1906 by [[Maurice Fréchet]], though it was Hausdorff who popularised the term "metric space" ({{langx |de| metrischer Raum}}).<ref>
{{oed | metric space}}
</ref><ref>
{{cite book
 |last1                = Hausdorff
 |first1               = Felix
 |author-link1         = Felix Hausdorff
 |orig-date            = 1914
 |chapter              = Punktmengen in allgemeinen Räumen
 |title                = Grundzüge der Mengenlehre
 |url                  = https://books.google.com/books?id=xMZXAAAAYAAJ
 |series               = Göschens Lehrbücherei/Gruppe I: Reine und Angewandte Mathematik Serie
 |date = 1914
 |language             = de
 |location             = Leipzig
 |publisher            = Von Veit
 |publication-date     = 2011
 |page                 = 211
 |isbn                 = 9783110989854
 |access-date          = 20 August 2022
 |quote                = Unter einem m e t r i s c h e n &nbsp; R a u m e verstehen wir eine Menge ''E'', [...].
}} 
</ref>{{better source|date=June 2024}}