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Normal spaces and the Lusin-Menchoff property. (English) Zbl 0897.54001
Let \((X,\rho)\) be a topological space. According to reviewer’s terminology [Commentat. Math. Univ. Carol. 18, 515-530 (1977; Zbl 0359.54013)], another topology \(\tau \) on \(X\) finer than \(\rho \) is said to have the Lusin-Menchoff property if for each pair of disjoint subsets \(F_{\rho }\) and \(F_{\tau }\) of \(X\), \(F_{\rho }\) \(\rho \)-closed, \(F_{\tau }\) \(\tau \)-closed, there are disjoint sets \(G_{\rho }\) and \(G_{\tau }\), \(G_{\rho }\) \(\rho \)-open, \(G_{\tau }\) \(\tau \)-open such that \(F_{\rho } \subset G_{\tau }\) and \(F_{\tau } \subset G_{\rho }\). The author studies the relation between the Lusin-Menchoff property of a “fine” topology \(\tau \) and the normality of \(\tau \) using the so-called \(F_{\sigma }\)-separation property. He introduced three interesting counterexamples labelled as the train topology, the cuckoo topology and the jump topology.
Reviewer: J.LukeŇ° (Praha)
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
26A03 Foundations: limits and generalizations, elementary topology of the line
31C40 Fine potential theory; fine properties of sets and functions
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