Cosheaves and connectedness in formal topology

The localic definitions of cosheaves and connectedness are transferred from impredicative topos theory to predicative formal topology. A formal topology is locally connected (has base of connected opens) iff it has a cosheaf π0 together with certain additional structure and properties that constrain π0 to be the connected components cosheaf. In the inductively generated case, complete spreads (in the sense of Bunge and Funk) corresponding to cosheaves are defined as formal topologies. Maps between the complete spreads are equivalent to homomorphisms between the cosheaves. For any cosheaf, the following are equivalent: (i) it is the connected components cosheaf for a locally connected formal topology; (ii) its complete spread is a homeomorphism; (iii) it is “strongly terminal”.