As we have remarked, there is a natural topology for a finite-dimensional real linear space X, that induced by any norm on X. It is a fair supposition that there should be more or less natural topologies also for the classical groups, Spin groups, Grassmannians and quadric Grassmannians, all of which are closely related to finite-dimensional linear spaces. It turns out that they also all have natural smooth structures as well, the groups being examples of Lie groups.

Important topological properties of the classical groups are their compactness or connectedness or otherwise.

Topological groups

A topological group consists of a group G and a topology for G such that the maps

G × G → G; (a, b) ↦ ab and G → G; a ↦ a–1

are continuous. An equivalent condition is that the map G × G; (a, b) ↦ a–1b is continuous.

Examples 22.1Let X be a finite-dimensional real linear space. Then the group GL(X) is a topological group.

Topological group maps and topological subgroups are defined in the obvious ways.

Proposition 22.2Any subgroup of a topological group is a topological group.

Corollary 22.3All the groups listed in Table 13.10 are topological groups.

Proposition 22.4For any p, q the group Spin(p, q), regarded as a subgroup of the Clifford algebra Rp,q is a topological group and the map

defined in Proposition 16.14 is a topological group map.