While recently attempting to compile a glossary of terms related to geometric algebra, I found myself in several places wanting to use the term "geometric space". Finally, of course, it too needed to be defined: at which point it became embarrassingly clear that I am unable to do so. The spaces/ symmetry groups with which GA is principally concerned are pretty standard: e.g. Euclidean, inversive/ conformal/ Moebius, spherical/ orthogonal, equilong/ Laguerre, contact/ Lie-sphere, 3-space line geometry. The constraints of the formalism actually restrict these to "quadratic space" --- but what I have in mind is more general than that --- for instance, geometry of lines in n-space (involving multiple quadratics). Conventional classifications based on topology, such as "metric space" --- as Dan Asimov recently pointed out --- rely on point-based notions which simply fail to apply in most of the cases above, even the familiar inversive geometry, because the "metric" (a quadratic form) is indefinite. Geometry of points on a real manifold bears some relation to what's required: but that relies on too much prior machinery for comfort. And the mind-numbingly abstract approach of "schemes", "stacks" and their like opens a whole counter-intuitive can of worms, which I have no intention of attempting to broach. So, I need a helping hand here, folks --- can somebody please come up with simple, elementary, intuitively plausible definition for what I am trying to discuss here? Fred Lunnon