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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. High Quality Content by WIKIPEDIA articles! In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group, G, in question is the homeomorphism group of the space, X. In this case X is homogeneous if intuitively X looks locally the same everywhere. Some authors insist that the action of G be effective (i.e. faithful), although the present article does not. Thus there is a group action of G on X which can be thought of as preserving some geometric structure on X, and making X into a single G-orbit.

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