Re: Stupid question: what space was Euclid working in? (almost)

tporter@informatics.bangor.ac.uk wrote:

If one assumes that `region' is a more basic notion of position than `point' a lot of Euclid still goes through but dimension seems very hard to handle.

The topological form of Helly's theorem might be a place to start. However, defining dimension in terms of _convex_ structure is very tricky once you get into general spaces. For instance, I showed in my thesis (in a section eventualLy rewritten for _Cahiers_) that if you define a "convex set" on S^1 to be an arc shorter than an open semicircle, you get the obvious homology. However, if closed semicircles and their intersections are convex, the homology becomes that of the 2-sphere. -Robert