An application of Mary Ellen Rudin's solution to Nikiel's conjecture
Following H. Junnila, a regular \(T_1\)-space \(X\) is called utterly normal if it has a magnetic base system, i.e., a collection \(\{{\mathcal B}_x\mid x\in X\}\), where each \({\mathcal B}_x\) is a base for the neighborhoods of \(x\), such that if \(B_x\in{\mathcal B}_x\) and \(B_y\in{\mathcal B}_y\) and \(B_x\cap B_y\neq\emptyset\), then either \(x\in\text{cl\,}B_y\) or \(y\in\text{cl\,}B_x\). Every utterly normal space is monotonically normal in the sense of \textit{R. W. Heath} et al. [Trans. Am. Math. Soc. 178, 481--493 (1973; Zbl 0269.54009)]. Whether the converse is true, i.e., whether every monotonically normal space is utterly normal, is unknown. It is also an open problem whether every utterly normal space has an open magnetic base system, i.e., a magnetic base system consisting of open sets. In light of these questions the main results of this paper are particularly interesting. It is first shown that every suborderable space is an utterly normal space with an open magnetic base system. Since, by M. E. Rudin's solution to Nikiel's conjecture, every compact monotonically normal space is the continuous image of a compact linearly ordered space, it follows that every compact monotonically normal space is utterly normal with an open magnetic base system. This fact is then used to prove that every compact monotonically normal space is hereditarily weakly orthocompact in the sense of \textit{B. M. Scott} [Stud. Topol., Proc. Conf. Charlotte, N. C., 1974, 517--537 (1975; Zbl 0311.54023)]. It follows that the one point compactification of a locally compact monotonically normal space \(X\) is monotonically normal if and only if \(X\) is weakly orthocompact. Hence every locally compact, weakly orthocompact, monotonically normal space is utterly normal with an open magnetic base system. It is unknown whether every stratifiable space is utterly normal, but the authors prove that every stratifiable space with an open magnetic base system is an \(M_1\) space.
scientific article; zbMATH DE number 6506268