Dear John, For point-set results like this it can be a bit delicate working out how the point-free topos treatment goes. Moerdijk has proved that for a connected, locally connected topos X, the map ends: X^I -> XxX is an open surjection. (Here I = [0,1] is the closed real interval, and if p: I -> X then ends(p) = (p(0), p(1)).) This is interpreted as the appropriate point-free way to say that X is path-connected, so connected, locally connected => path connected - which goes against the classical account. Part of the issue is that a point-free surjection is not necessarily surjective on points. Hence even for locally connected spaces, which are supposed to be the well behaved ones, the path-connected components got from the topos theory (which, by Moerdijk's result, agree with the connected components) may be different from the ones got from point-set topology. All the best, Steve. On 04/02/2018 20:57, John Baez wrote:
Steve Vickers wrote:
We get Stone spaces of connected components more generally for any compact regular space - take the Stone space corresponding to the Boolean algebra of clopens. People tend not to notice the Stone space aspects in the usual examples based on real analysis, since they are also locally connected. Being a Stone space then just makes the set of connected components finite with decidable equality. For any compact regular space, we find that each closed subspace has a Stone space of connected components.
Digressing a bit, this reminds me of some results David Roberts recently pointed out. However, they concern path-connected components rather than connected components. The set of path-connected components of a space X is a quotient set of X, so we can give it the quotient topology. What can the resulting space be like?
Anything! For every topological space X, there is a paracompact Hausdorff space whose space of path-connected components is homeomorphic to X.
D. Harris, Every space is a path-component space, Pacific J. Math. 91 no. 1 (1980) 95-104. http://dx.doi.org/10.2140/pjm.1980.91.95
There is more here:
https://mathoverflow.net/questions/291443/paths-in-path-component-spaces
Best, jb
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]