J. Alexander's 1926 paper, Combinatorial Analysis Situs, doesn't refer to the field as combinatorial topology.

He mentions that combinatorial analysis situs is concerned with topological invariants and topological constants.

However, he refers to the subject of the paper itself as a subfield of analysis situs, considered as a branch of geometry.

When/why did we transition from calling "the study of topological invariants" by the name "analysis situs" to calling it by the name "topology"?

  • $\begingroup$ German influence $\endgroup$ Aug 28 '15 at 1:13
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    $\begingroup$ I suspect this was partly due to the fact that around the mid 1920s the evolving generalizations of point set theory (Cantor, Borel, Frechet, Hausdorff, etc.) was becoming known as "general topology". Some discussion of the origin of the term is given at the bottom of p. 290 to the top of p. 291 of Chittenden, On general topology and the relation of the properties of the class of all continuous functions to the properties of space Transactions of the American Mathematical Society 31 #2 (April 1929), 290-321. $\endgroup$ Aug 31 '15 at 19:17
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    $\begingroup$ For more evidence that the term general topology was "in the air" in the mid to late 1920s, there is also Sierpinski's 1928 book Topologia Ogólna, the book that Cypra Cecilia Krieger published an English translation of in 1934 with the title Introduction to General Topology. $\endgroup$ Aug 31 '15 at 19:20

According to MacTutor, "the subject was known as analysis situs for many years and only in the late 1920s was the English word topology used by Lefschetz". Lefschetz's 1924 work is titled Analysis Situs and Algebraic Geometry, but his influential 1930 monograph is already Topology, which he defines as "doctrine of the modal features of objects, or of the laws of connection, of relative position and of succession of points, lines, surfaces, bodies and their parts, or aggregates in space, always without regard to matters of measure or quantity..".

"Why" is speculative. From an answer on MSE:"Lefshetz [Topology, Amer. Math. Soc. Colloq. Publ 12 (1930), page 361] wrote that Poincaré had tried to develop the subject along `analytic' lines, but had turned instead to combinatorial methods because the analytic approach failed for example in the Poincaré duality theorem". Since the focus shifted from analytic to combinatorial methods "analysis" was not a fitting name. Listing published a book Vorstudien zur Topologie in 1847 and used the word since 1836. According to his 1883 obituary it was meant to distinguish "...qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated", which clearly resonates with Lefschetz's description. So it was a natural choice to make a point. Later, a similar change in perspective was reflected in the switch from "combinatorial topology" to "algebraic topology", attributed to Hopf, under Noether's influence, Mayer and Vietoris.


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