Deitmar Anton - Principles of Harmonic Analysis

Any given group becomes a topological group when equipped with the discrete topology , i.e., the topology, in which every subset is open. In this case we speak of a discrete group

The additive and multiplicative groups and of the field of real numbers are topological groups with their usual topologies. So is the group of all real invertible matrices, which inherits the -topology from the inclusion , where denotes the space of all matrices over the reals. As for the proofs of these statements, recall that in analysis one proves that if the sequences a i and b i converge to a and b , respectively, then their difference converges to ab , and this implies that is a topological group. The proof for the multiplicative group is similar. For the matrix groups recall that matrix multiplication is a polynomial map in the entries of the matrices, and hence continuous. The determinant map also is a polynomial and so the inversion of matrices is given by rational maps, as for an invertible matrix A one has , where is the adjugate matrix of A ; entries of the latter are determinants of sub-matrices of A , therefore the map is indeed continuous.