User:Kompik/Math/Filterhistory

This is an auxiliary page concerning the history of the notion of filter. Feel free to add additional references if you stumble upon any of them.

Some authors mention that filter base was used by L. Vietoris [V] under the name “Kräntze” prior to H. Cartan ([P, p.46], [R1],[R2]). R. M. Dudley [D, p.76] mentions that C. Caratheodory used filter bases in [C] and M. H. Stone was in fact dealing with filters in [S].

[C] C. Caratheodory. ¨Uber die Begrenzung einfach zusammenhängender Gebiete. Math. Ann., 73:323–370, 1913.

[D] R. M. Dudley. Real Analysis and Probabilty. Cambridge University Press, Cambridge, 2002.

[P] G. Preuss. Foundations of General Topology. Kluwer, Dordrecht, 2002.

[R1] H. Reitberger. The contributions of L. Vietoris and H. Tietze to the foundations of general topology. In C. E. Aull and R. Lowen, editors, Handbook of the history of general topology, Volume 1, pages 31–40. Kluwer, Dordrecht, 1997.

[R2] H. Reitberger. Leopold Vietoris (1891–-2002), Notices of the American Mathematical Society} 49 (10):1231--1236, 2002. [1]

[S] Marshall Harvey Stone. The theory of representations for Boolean algebras. Trans. Amer. Math. Soc., 40:37–111, 1936.

[V] L. Vietoris. Stetige Mengen. Monatshefte f. Math., 31:545–555, 1921.

Additional references

According to Engelking "The setting of convergence in terms of filters was sketched by Cartan in [1937] amd [1937a], and was developed by Bourbaki in [1940] (filters first appeared in Riesz [1908])". See Engelking, General topology, revised edition, 1989, page 55 (unfortunately, no web link)
According to Comfort "Ultrafilters were apparently first defined, and shown to exist (on omega), by F. Riesz [Ri] and Ulam [UI], respectively". [2].
I cannot find a copy of Riesz paper. Paolo Lipparini (talk) 18:38, 27 September 2011 (UTC)