|WikiProject Mathematics||(Rated C-class, High-importance)|
I've removed scheme (mathematics) from the introduction. It isn't a straightforward thing to explain what it means for a group action on a scheme S to be 'transitive'; nor what the 'orbits' of such an action are. It would be better to make it algebraic variety; but in any case one can regard that as a special case of a topological space without too much damage.
Charles Matthews 10:47, 21 Aug 2004 (UTC)
Homogeneous space without reference to a group
I've heard the definition of "homogeneous space" without reference to a group or group actions (this was in an introductory topology course), namely a space in which, for any points x, y in X, there exists a homeomorphism f from X to itself such that f(x) = y. Since the homeomorphisms are a group, this is just the action on X by evaluation which is transitive if it satisfies the preceding property. I think (but don't know for sure!) that this definition is pretty common (I'm guessing this from the entry on the Cantor set), so I inserted the "If X is simply called a homogeneous space without reference to a group, it is usually assumed that..." I don't know precisely how usually though... so anyone who knows better can edit away!
Choni 18:34, 27 Aug 2004 (UTC)
Requirement of continuity
The requirement of continuity seems to me to be overly restrictive: a restriction of convenience for those not interested in other cases, but (or so it would seem to me) to be of no direct import to the concept of a homogeneous space.
An example Where the concept is of interest but where continuity does not apply is with geometries over finite fields. Essentially, the concept is one which says that every point of the space is indistinguishable from every other point: the group of symmetries maps every point to every point in the geometry. Thus, transitivity of the group action is the only requirement. I'm stripping out the continuity requirement in the lead, but feel free to correct me if I did it wrong. —Quondum 00:43, 18 September 2014 (UTC)