Monotone normality and neighborhood assignments
This paper largely deals with the proof of two Theorems, i.e., 1.1 and 1.2 and the construction of an important example, i.e., Example 3.4. The motivation for the paper are questions posed in [\textit{C. R. Borges} and \textit{A. C. Wehrly}, Topol. Proc. 16, 7--15 (1991; Zbl 0787.54023)] and [\textit{R. Z. Buzyakova, V. V. Tkachuk} and \textit{R. G. Wilson}, Commentat. Math. Univ. Carol. 48, No. 4, 689--697 (2007; Zbl 1199.54141)]. In Theorem 1.1, the authors prove that every neighborhood assignment for a monotonically normal space has a kernel which is homeomorphic to some subspace of an ordinal. As a corollary, the authors show that every monotone neighborhood assignment for a monotonically normal space has a discrete kernel (see Corollary 1.3). In Example 3.4, the authors show that it is not true that every neighborhood assignment for an arbitrary given topological space \(X\) has a kernel which is homeomorphic to some subspace of an ordinal. Indeed there is a regular space which has a neighborhood assignment with no kernels homeomorphic to any subspace of an ordinal. But the authors mention from reference [loc. cit.] that every neighborhood assignment for an arbitrary given topological space \(X\) has a scattered kernel. This is indeed a partial answer to the question whether every monotonically normal space is dually discrete (see [loc. cit.]). The authors also prove Theorem 1.2 which states that for every neighborhood assignment \(N\) of a monotonically normal space \(X\) with a monotone normality operator \(H\), there is a non-empty subspace \(D\) of \(X\) such that \(D\) is homeomorphic to some subspace of an ordinal and \(H(x, H(x, N(x))) \cap D = \emptyset\) for every \(x \in X \backslash N[D]\).
scientific article; zbMATH DE number 6023675
scientific article