Lower and upper regularizations of frame semicontinuous real functions
The main gist of this paper is a characterization of extremally disconnected frames (also known as De Morgan frames) in terms of extendibility of bounded real-valued continuous functions on open sublocales. The main theorem states that a frame \(L\) is extremally disconnected if and only if, for every open sublocale \(S\), every bounded real-valued continuous function on \(S\) extends to a bounded real-valued continuous function on \(L\). The authors establish this result by first defining the pointfree concepts of lower and upper regularizations of frame semicontinuous real-valued functions. The stated theorem is a much strengthened version of an earlier effort in this regard by \textit{Y.-M. Li} and \textit{Z.-H. Li} [``Constructive insertion theorems and extension theorems on extremally disconnected frames'', Algebra Univers. 44, No. 3--4, 271--281 (2000; Zbl 1013.06010)], which was demonstrated by \textit{J. Picado} [``A new look at localic interpolation theorems'', Topology Appl. 153, No. 16, 3203--3218 (2006; Zbl 1104.06007)] to have shortcomings emanating from a treatment of pointfree semicontinuity which was not fully reflective of its point-sensitive counterpart. The preparatory results in the present paper are a continuation of the authors' study of semicontinuity in pointfree topology.
scientific article; zbMATH DE number 5652276