Homogeneous space

In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. That is, there is a group action of G on X, respecting the given geometric structure of X, and making X into a single G-orbit. (It is assumed, therefore, that X isn't empty.) If X is simply called a homogeneous space without reference to a group, it is usually assumed that G is the group of all homeomorphisms from X to itself (the automorphism group of X), with the natural action of evaluation.

See also

• properly discontinuous
• free regular set