Pointfree forms of Dowker's and Michael's insertion theorems
The authors prove two strict insertion theorems for frame homomorphisms. When applied to the frame of all open subsets of a topological space, they are equivalent to the following insertion theorems of Dowker and Michael regarding, respectively, normal countably paracompact spaces and perfectly normal spaces. (Dowker) A topological space \(X\) is normal and countably paracompact if and only if, given \(h,g:X\rightarrow {\mathbb R}\) such that \(h<g, h\) is upper semicontinuous and \(g\) is lower semicontinuous, there is a continuous \(f:X\rightarrow {\mathbb R}\) such that \(h<f<g.\) (Michael) A topological space \(X\) is perfectly normal if and only if, given \(h,g:X\rightarrow {\mathbb R}\) such that \(h\leq g, h\) is upper semicontinuous and \(g\) is lower semicontinuous, there is a continuous \(f:X\rightarrow {\mathbb R}\) such that \(h\leq f\leq g\) and \(h(x)<f(x)<g(x)\) whenever \(h(x)<g(x).\) The authors state that, however, in the pointfree context they have been unable to provide pointfree assertions corresponding exactly to these two classical insertion results. Furthermore, they present a study of the concept of perfect normality for frames.
scientific article; zbMATH DE number 5364859
scientific article