# History of various definitions of topology

I have been reading Point Set Topology for a while, and turns out that there are various possible ways to define a topology. Most popular one is using open set axioms. Another one is using closure axioms, which was introduced by Kuratowski.

I am interested to know name of the researchers who had given the definition of topological spaces in the following ways:

1. Definition through Open sets (The most popular one)
2. Definition through neighbourhood system
3. Definition through interior points

• Definition 1 was popularized by N. Bourbaki and J. Kelley, the author of one of the first English books on general topology. – Alexandre Eremenko Mar 24 '19 at 12:08
• There is also a definition via convergence of nets (generalization of sequences, the term was coined by Kelley) due to Moore-Smith, which goes back to Cartan's notion of filters. – Conifold Mar 25 '19 at 6:10

By the time these definitions were introduced as definitions there was a body of previous work, where they were convenient side notions for stating theorems in special cases for subsets of real line, plane, and then curve and function spaces (Cantor's accumulation points, derived and closed sets on the line, Weierstrass's theorems, Heine-Borel, Ascoli-Arzelà, Frechet's metric notions, etc.). The first axiomatic definitions are due to Riesz (accumulation points, 1906) and Hausdorff (neighborhoods, 1914). Hausdorff's Grundzüge der Mengenlehre became a model of axiomatic exposition of both set theory and topology. Some details from Kreyszig's chapter in the Handbook of History of General Topology, vol.1:

"The earliest definition of a topological space was given in 1906 (and somewhat obscurely published in 1907) by F. Riesz (1880-1956), with an abstract in the Atti of the International Congress of Mathematicians in Rome in 1908 (cf. , I, 110-154, 155-161). Of the possible concepts to be axiomatized (neighborhood, closure, etc.), Riesz chose accumulation point, as his axioms will show. He followed Hilbert in calling it Verdichtungsstelle (condensation point, which is now used in a different sense).

...It is not known why Riesz did not make neighborhood his basic concept to be axiomatized (as in Hausdorff's definition of topological space). It may be that he wished to stay more closely to analysis rather than to geometry. To fully understand this, we should recall that the concept of a (general) neighborhood of a point developed only slowly, even in the plane, where it was first used as late as in 1902, by Hilbert in a paper on the foundations of geometry (Math. Ann. 56, 381-422). Neighborhoods ,also occurred later in work by Hedrick and by Frechet, mentioned in , 1020...

In his classic Grundzüge der Mengenlehre , Hausdorff created a useful concept of a topological space by axiomatizing 'neighborhood'. His well-known axioms included his separation axiom T2 (that any two distinct points have disjoint neighborhoods), so that he obtained a Hausdorff space. Working out his theory of topological and metric spaces abstractly but with applicability in mind, he made his book a landmark that has often been regarded as the beginning of set-theoretic topology as we understand it today.

Subsequently, multiple alternative definitions appeared, which are referenced in Kelley's General Topology (Notes to Chapter 1), starting with the volumes of Fundamenta Mathematica following the publication of Hausdorff's book in 1914 and going into 1940-s. I am not sure which one is the "interior point" one. The open set definition appears in Bourbaki's General Topology (1940). They do not credit anyone for it, nor does Kelley, but then it is closely related to the neighborhood version. Their short surmise of history appears in the Historical Note to Chapter 1:

"The first attempts to abstract what is common to properties of sets of points and sets of functions are due to Frechet  and F. Riesz . The former started from the notion of countable limit and did not succeed in constructing a convenient and fruitful system of axioms...

General topology as it is understood today began with Hausdorff (, Chapters 7, 8, 9), who again took up the concept of neighbourhood (by which he meant what in the terminology of this series of volumes is called an "open neighbourhood") and chose from Hilbert's axioms for neighbourhoods in the plane those which gave his theory all the precision and generality desired. The axioms he took as a starting-point were essentially (taking into account the difference between his concept of neighbourhood and ours) axioms (V$$_I$$), (V$$_{II}$$), (V$$_{III}$$), (V$$_{IV}$$) of § 1 and (H) of § 8, and the chapter in which he develops the consequences of these axioms has remained a model of axiomatic theory, abstract but adapted in advance to applications.

...Finally, the introduction of filters by H. Cartan  has brought to topology a valuable instrument, usable in all sorts of applications (in which it replaces to advantage the notion of "Moore-Smith convergence" ). Furthermore, the development of the theorem on ultrafilters (Theorem I, § 6), has clarified and simplified the theory."

Cartan essentially rediscovered what was done by Vietoris in 1913-19.

Definition through interior points can be obtained by its dual version first proposed by Kuratowski in 1922, which is known as Kuratowski Closure Algebra. Historically, definition through closure points (Closure Algebra) came first and the one by interior points followed after it.

The following three versions for the definition of topological spaces are equivalent:

• The neighbourhood system by Hausdorff in his famous work "Grundzüge der Mengenlehre" in 1914.
• The Closure Algebra by Kuratowski from 1922.
• The current used system through Open sets by Bourbaki in "Topologie" in 1940.