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
Steve Vickers wrote: 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/ ]