Inventor of Topological groups

Question

I had difficulty in finding the person who introduced the term "topological groups". I just want to know who introduced the term topological groups.

Answer 1

According to a 1945 article of Montgomery in The American Mathematical Monthly titled "What is a Topological Group?" the idea of topological groups started with Sophus Lie. See here: https://www.jstor.org/stable/2305290?seq=1#metadata_info_tab_contents

Answer 2

Topological spaces in their modern sense were invented by Hausdorff in 1914, while Lie died in 1899. So the notion has certainly been invented after 1914 (although Lie deserves much credit in the development of these ideas!). The earliest I can detect is: F. Leja, sur la notion de groupe abstrait topologique, Fund Math (1927) 9(1), 37-44 (French). (EUDML, freely available)

See also this historical survey, in French, by J-P. Pier: http://www.numdam.org/article/CSHM_1988__9__1_0.pdf

Note: Beware that I found "topological group" used by Alexander in 1922 in the modern meaning of "fundamental group".]

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Added (Sep. 2023): at the end of the Bourbaki chapter on topological group, in the short "historical note" I can read (my translation): "The study of general topological groups was inaugurated by O. Schreier in 1926 (I)"

(I) O. Schreier. Abstrakte kontinuierliche Gruppen, Abh. Math. Seminar. d. Hamburger Univ., t. IV 1926, p15-32.

The above Feja 1927 paper actually quotes, p2, this 1926 Schreier paper without detail (just saying "see also [reference]").