Monotone insertion of lattice-valued functions

Abstract

Insertion of lattice-valued functions in a monotone manner is investigated. For L a -separable completely distributive lattice (i.e. L admits a countable base which is free of supercompact elements), a monotone version of the Kat¥tovTong insertion theorem for L-valued functions is established. We also provide a monotone lattice-valued version of Urysohn's lemma. Both results yield new characterizations of monotonically normal spaces. Moreover, extension of lattice-valued functions under additional assumptions is shown to characterize also monotone normality.

Monotone normality free of T 1 axiom

“…Our nal result extends Theorem 2.3 in [9] to the T 1 -free context. That theorem was a generalization to lattice-valued functions of the extension theorem for monotonically normal spaces given by Stares in [11,Theorem 2.3].…”

“…Note that it extends Proposition 3.3 and improves Proposition 3.7 in [9], since L is now only assumed to be completely distributive, instead of being also required to have a countable base. For the case of real-valued functions, the equivalence (1) ⇔ (3) in the T 1 -free context was obtained in [8].…”

“…Among the dierent possibilities to dene semicontinuity for latticevalued functions, in this paper, as in [6] and [9], we will work with the denition below:…”

“…Let us now recall a result from [9], which will be needed to prove the characterization of monotone normality in terms of extension of lattice-valued functions in the T 1 -free context. Note that for the case of real-valued functions, the result below was given by Kubiak in [8].…”

“…This idea is as simple as eective. It is also used in Section 4, to provide an extension property of lattice-valued functions for monotonically normal spaces, extending the result given in [9] to a T 1 -free context (see also [11] for the case of real valued functions).…”