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