## Abstract

A *topological group* is a set *G* that has the structure of a group (say with group operation (*x*,*y*) → *xy*) and of a topological space, and is such that the operations (*x*,*y*) → *xy* and *x* → *x* ^{−1} are continuous. Note that (*x*,*y*) → *xy* is a function from the product space *G* × *G* to *G*, and that we are requiring that it be continuous with respect to the product topology on *G* × *G*; thus *xy* must be “jointly continuous” in *x* and *y*, and not merely continuous in *x* with *y* held fixed and continuous in *y* with *x* held fixed (see Exercise 3). A *locally compact topological group*, or simply a *locally compact group*, is a topological group whose topology is locally compact and Hausdorff. A *compact group* is a topological group whose topology is compact and Hausdorff.

## Keywords

Open Subset Compact Subset Open Neighborhood Topological Group Compact Group## Preview

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## Notes

- The history of Haar measure is summarized in the notes at the ends of Sections 15 and 16 of Hewitt and Ross [41].Google Scholar
- The reader can find a more extensive introduction to topological groups in Pontryagin [69] or in Hewitt and Ross [41].Google Scholar
- The proof given here for the existence of Haar measure (which is a modification of Halmos’s modification of Weil’s [86] proof) depends on the axiom of choice. Proofs that do not depend on this axiom have been given by Cartan [16] and by Bredon [12], Cartan’s proof is given by Hewitt and Ross [41] and by Nachbin [65]. Hewitt and Ross and Nachbin also give calculations of Haar measure for a number of groups.Google Scholar