The connected Vietoris powerlocale

The Vietoris powerlocale V X is a point-free analogue of the Vietoris hyperspace. In this paper we introduce and study a sublocale V X whose points are those points of V X that (considered as sublocales of X) satisfy a constructively strong connectedness property. V c is a strong monad on the category of locales. The strength gives rise to a product map × : V X × V Y → V (X × Y ), showing that the product of two of these connected sublocales is again connected. If X is locally connected then V X is overt. In the case where X is the localic completion Y of a generalized metric space Y , the points of V Y are characterized as certain Cauchy filters of formal balls for the finite power set FY with respect to a Vietoris metric. The results are applied to the particular case of the point-free real line R, giving a choice-free constructive version of the Intermediate Value Theorem and Rolle’s Theorem. The work is constructive in the sense of topos-validity with natural numbers object. Its geometric aspects (preserved under inverse image functors) are stressed, and exploited to give a pointwise development of the point-free locale theory. The connected Vietoris powerlocale itself is a geometric construction.