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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.Lukeš (Praha)
##### MSC:
 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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