nLab topology -- references

References

Historical origins

The general idea of topology goes back to:

• Henri Poincaré, Analysis Situs, Journal de l’École Polytechnique 2 1 (1895) 1–123 [gallica:12148/bpt6k4337198/f7, Engl: pdf, pdf]

The notion of topological space involving neighbourhoods was first developed, for the special case now known as Hausdorff spaces, in:

• Felix Hausdorff, Grundzüge der Mengenlehre, Leipzig: Veit (1914), Reprinted by Chelsea Publishing Company (1944, 1949, 1965) [ISBN:978-0-8284-0061-9, ark:/13960/t2891gn8g]

The more general definition – dropping Hausdorff’s $T_2$-separation axiom and formulated in terms of closure operators that preserve finite unions – is due to:

• Kazimierz Kuratowski, Sur l’opération Ā de l’Analysis Situs, Fundamenta Mathematicae 3 (1922) 182–199 [doi:10.4064/fm-3-1-182-199]

The modern formulation via open set was widely popularized by:

• Nicolas Bourbaki], Eléments de mathématique II. Première partie. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitre I. Structures topologiques. Actualités scientifiques et industrielles, vol. 858. Hermann, Paris (1940)

  General topology, Elements of Mathematics III, Springer (1971, 1990, 1995) [doi:10.1007/978-3-642-61701-0]

Further

Further textbook accounts:

• John Kelley, General topology, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [ISBN:978-0-387-90125-1]

• James Dugundji, Topology, Allyn and Bacon 1966 (pdf)

• James Munkres, Topology, Prentice Hall (1975, 2000) [ISBN:0-13-181629-2, pdf]

• Richard E. Hodel (ed.), Set-Theoretic Topology, Academic Press (1977) [doi:10.1016/C2013-0-11355-4]

• Klaus Jänich, Topology, Undergraduate Texts in Mathematics, Springer (1984, 1999) [ISBN:9780387908922, doi:10.1007/978-3-662-10574-0, Chapters 1-2: pdf]

• Ryszard Engelking, General Topology, Sigma series in pure mathematics 6, Heldermann 1989 (ISBN 388538-006-4)

• Steven Vickers, Topology via Logic, Cambridge University Press (1989) (toc pdf)

• Glen Bredon, Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)

• Terry Lawson, Topology: A Geometric Approach, Oxford University Press (2003) (pdf)

• Anatole Katok, Alexey Sossinsky, Introduction to Modern Topology and Geometry (2010) [toc pdf, pdf]

and leading over to homotopy theory:

• Ioan Mackenzie James, General Topology and Homotopy Theory, Springer 1984

  (doi:10.1007/978-1-4613-8283-6)

On counterexamples in topology:

• Lynn Arthur Steen, J. Arthur Seebach, Counterexamples in Topology, Springer 1971 (doi:10.1007/978-1-4612-6290-9)

With emphasis on category theoretic aspects of general topology, notably on $T_n$-reflections:

• Horst Herrlich, George Strecker, Categorical topology – Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971 (pdf), pages 255-341 in: C. E. Aull, R Lowen (eds.), Handbook of the History of General Topology. Vol. 1, Kluwer 1997 (doi:10.1007/978-94-017-0468-7)

• Tai-Danae Bradley, Tyler Bryson, John Terilla, Topology – A categorical approach, MIT Press 2020 (ISBN:9780262539357, web version)

See also:

• Alan Hatcher, Algebraic Topology

• Neil Strickland, A Bestiary of Topological Objects [pdf, pdf]

and see further references at algebraic topology.

Lecture notes:

• Friedhelm Waldhausen, Topologie (pdf)

• Alex Kuronya, Introduction to topology, 2010 (pdf)

• Anatole Katok, Alexey Sossinsky, Introduction to modern topology and geometry (pdf)

• Urs Schreiber, Introduction to Topology, Bonn 2017

• Michael Müger, Topology for the working mathematician, Nijmegen 2018 (pdf)

Basic topology set up in intuitionistic mathematics is discussed in

• Franka Waaldijk, modern intuitionistic topology, 1996 (pdf)

See also:

• Topospaces, a Wiki with basic material on topology.