Let be a topological space on which a topological group acts continuously and freely. The map is called a principal bundle (or principal -bundle) if the projection map is a locally trivial bundle.

Any principal bundle with a section is trivial, since the map given by is an isomorphism. In particular, any -bundle which is topologically trivial is also isomorphic to as a -space. Thus any local trivialization of as a topological bundle is an equivariant trivialization.