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

["Robert J. MacG. Dawson" <rdawson@cs.stmarys.ca>]

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