Monotone insertion of continuous functions
Amongst classical topological insertion theorems are those of Katětov-Tong (characterizing normal spaces), Dowker (characterizing normal countably paracompact spaces), and Michael (characterizing perfectly normal spaces). The monotonization of the Katětov-Tong theorem has been shown by the reviewer [Quest. Answers Gen. Topology 11, No. 1, 51-59 (1993; Zbl 0826.54023)] to characterize monotonically normal spaces. A monotone version of Michael's theorem is a characterization of stratifiable spaces (as attributed to \textit{P. Nyikos} and \textit{C. Pan}, to appear). The main result of the paper under review is that the monotone version of the Dowker insertion theorem also does characterize stratifiable spaces. More specifically, for \(X\) a topological \(T_1\)-space, let \(UL_<(X) =\{(g,h)\in USC(X)\times LSC(X):g<h\) on \(X\}\) be endowed with the componentwise order, i.e., \((g_1,h_1) \leq(g_2,h_2)\) iff \(g_1\leq g_2\) and \(h_1\leq h_2\). The authors prove Theorem 1.6: \(X\) is stratifiable if and only if there exists an order-preserving mapping \(\Phi: UL_<(X)\to C(X)\) such that \(g<\Phi (f,g)<h\) for all \((g,h)\in UL_<(X)\). The authors also prove monotone versions of insertion theorems of \textit{R. L. Blair} and \textit{M. A. Swardson} [Indian J. Math. 29, No. 3, 229-250 (1987; Zbl 0666.54012)] for hedgehog-valued functions on normal spaces. These versions are characterizations of monotonically normal spaces, too. Other results include hedgehog-valued versions of the classical theorems of Dowker and Michael. Reviewer's note: A proof of the monotone version of the Michael insertion theorem has appeared in a very recent paper by \textit{E. Lane}, \textit{P. Nyikos}, and \textit{C. Pan} [Acta Math. Hung. 92, No. 3, 219-231 (2001)].
scientific article; zbMATH DE number 1538679