Re: Topos theory for spaces of connected components

[John Baez <baez@math.ucr.edu>]

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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