A new look at some classical theorems on continuous functions on normal spaces

Abstract

A sucient condition for the strict insertion of a continuous function between two comparable upper and lower semicontinuous functions on a normal space is given. Among immediate corollaries are the classical insertion theorems of Michael and Dowker. Our insertion lemma also provides purely topological proofs of some standard results on closed subsets of normal spaces which normally depend upon uniform convergence of series of continuous functions. We also establish a Tietze-type extension theorem characterizing closed G δ -sets in a normal space.

Pointfree forms of Dowker’s and Michael’s insertion theorems

“…In [13], there is an insertion lemma holding for normal topologies L = OX with U V iff U ⊆ V . It continues to hold for arbitrary frames with a strong Katětov relation:…”

“…It should however be emphasized that both Theorems B and C easily follow from their pointfree versions established in this paper. These versions are corollaries of a rather general insertion lemma related to an arbitrary frame L with a certain extra order which in turn is an abstract version of a result of Gutiérrez García and Kubiak [13] concerning a normal topology OX with U V iff int(X \ U) ∪ V = X . We also establish some natural results regarding perfectly normal frames.…”

Perfect locales and localic real functions

“…By Proposition 3.4 (1), ι(0, ϕ G ) # = o(U ) and hence it follows from Proposition 3.5 (3) and Corollary 3.6 (3) that U = F −1 (0, +∞) = (G−F ) −1 (0, +∞). Consequently, 0 < F | U < G| U .…”