Historical note

(Numbers in brackets refer to the bibliography at the end of this note.)

The general theory of topological groups is one of the most recent branches of analysis. However, particular topological groups were well known a long time ago; and in the second half of the nineteenth century Sophus Lie built up the vast theory of those topological groups which he called "continuous groups" and which are nowadays known as "Lie groups". The reader will find fuller information on the genesis and development of this theory in the Historical Notes attached to the volume on Lie groups in this series.

The study of general topological groups was initiated by O. Schreier in 1926 [1]. Since then it has been the subject of much work, which has, among other things, to a large extent elucidated the structure of locally compact groups. Here our intention has been to give only the most elementary definitions and results of the theory, and we refer the reader who wishes to go deeper into the subject to the monographs of L. Pontrjagin [2], A. Weil [3], and D. Montgomery and L. Zippin [4].

[1] O. Schreier, Abstrakte kontinuierliche Gruppen, Hamb. Abh. 4 (1926), p. 15. [2] L. Pontrjagin, Topologische Gruppen, 2 volumes, Leipzig (Teubner), 1957. [3] A. Weil, L'intégration dans les groupes topologiques et ses applications, Act. Sci. et Ind. no. 869, Paris (Hermann) 1940 [2nd edition, no. 869-1145, Paris (Hermann), 1953]. [4] D. Montgomery and L. Zippin, Topological Transformation Groups, New York (Interscience), 1955.