Completely normal frames and real-valued functions

Abstract

Up to now point-free insertion results have been obtained only for semicontinuous real functions. Notably, there is now available a setting for dealing with arbitrary, not necessarily (semi-)continuous, point-free real functions, due to Gutiérrez García, Kubiak and Picado, that gives point-free topology the freedom to deal with general real functions only available before to point-set topology. As a first example of the usefulness of that setting, we apply it to characterize completely normal frames in terms of an insertion result for general real functions. This characterization extends a well known classical result of T. Kubiak about completely normal spaces. In addition, characterizations of completely normal frames that extend results of H. Simmons for topological spaces are presented. In particular, it follows that complete normality is a lattice-invariant property of spaces, correcting an erroneous conclusion in [Y.-M. Wong, Lattice-invariant properties of topological spaces,

On hereditary properties of extremally disconnected frames and normal frames

“…In particular, for A = L we have A o(a) = o(a) and, therefore, both Theorem 3.7 of [11] and Proposition 3.3 of [14] follow: Corollary 5.3. The following are equivalent for any frame L:…”

“…Complete normality formulated in frames appeared for the first time in the literature with [19] (see also [31]), in the following form: a frame is completely normal if every pair S, T of separated sublocales of L (i.e. such that S ∩ T = {1} = S ∩ T) is separated by open sublocales, that is, there exist open sublocales U and V of L such that U ∩ V = {1}, S ⊆ U and T ⊆ V. As proved in [11,Proposition 3.3], this is equivalent to condition (CN) above.…”

“…A characterization of CN solely in terms of the lattice Ω(X) of all open sets of X is given in [30]: Proposition 1.2. ( [30,11]) The following are equivalent for any topological space X:…”

General insertion and extension theorems for localic real functions

“…After having introduced a localic analogue of the concept of an arbitrary (not necessarily continuous) real valued function (by Gutiérrez García et al [12]), we now have new areas of topology to be explored in the world of locales (cf. the very recent paper [8]). The motivation of this paper is the lack of localic variants of two topological results (the first of which involves arbitrary not necessarily continuous real valued function), viz.…”

On Property (A) and the socle of the $f$-ring $Frm(mathcal{P}(mathbb R), L)$

“…Let L be a frame and F (L) := F rm(L(R), SL), where SL is the dual of the co-frame of all sublocales of L. In [10], they showed that the lattice ordered ring F (L) is a pointfree counterpart of the ring R X with X a topological space (also see [9,11]). They thus have a pointfree analogue of the concept of an arbitrary, not necessarily (semi) continuous, real function on L. In [25], they showed that F (L) = C(S c (L)) is always order complete, where S c (L) is a frame of closedly generated sublocales.…”