Insertion of lattice-valued and hedgehog-valued functions

Abstract

Problems of inserting lattice-valued functions are investigated. We provide an analogue of the classical insertion theorem of Lane [Proc. Amer. Math. Soc. 49 (1975) 90–94] for L-valued functions where L is a ⊲-separable completely distributive lattice (i.e. L admits a countable join-dense subset which is free of completely join-irreducible elements). As a corollary we get an L-version of the Katětov–Tong insertion theorem due to Liu and Luo [Topology Appl. 45 (1992) 173–188] (our proof is different and much simpler). We show that ⊲-separable completely distributive lattices are closed under the formation of countable products. In particular, the Hilbert cube is a ⊲-separable completely distributive lattice and some join-dense subset is shown to be both order and topologically isomorphic to the hedgehog J ( ω ) with appropriately defined topology. This done, we deduce an insertion theorem for J ( ω ) -valued functions which is independent of that of Blair and Swardson [Indian J. Math. 29 (1987) 229–250]. Also, we provide an iff criterion for inserting a pair of semicontinuous function which yields, among others, a characterization of hereditarily normal spaces.