Re: Topos theory for spaces of connected components

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

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
>
>



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