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- W2081162393 abstract "We construct an absolute example of a space having the properties in the title. Let Y be the set of nonempty finite subsets of the Cantor cube of countable weight. The Pixley-Roy topology on Y is not normal, but the Vietoris topology on Y is normal. Our space can be considered a normalization of the Pixley-Roy topology on Y by adding cluster points which as a subspace have the Vietoris topology. The Alexandroff duplicating procedure is used liberally to glue the space together. The example is also a sigma compact paracompactp-space. If further set-theoretic assumptions are made (e.g. V = L or MA + CH), then it is known that even perfectly normal such examples exist. 0. Introduction. A space X is ccc if each collection of disjoint open sets is at most countable. A Souslin line is a ccc nonseparable linearly ordered topological space. It is consistent with the usual axioms for set theory (i.e. ZFC) that a Souslin line exists (Jech [7]). Such a line yields an example of a perfectly normal first countable ccc nonseparable space (cf. Rudin [11]). Martin's axiom plus the negation of the continuum hypothesis is also consistent with the usual axioms (Solovay and Tennenbaum [12]). It implies that there are no Souslin lines, nonetheless, one is still able to construct a perfectly normal first countable ccc nonseparable space (Przymusinski and Tall [10]). One is naturally led to wondering whether an absolute example with such properties can be constructed in ZFC alone. According to a theorem of F. Tall in [13], the existence of such a space implies either the existence of a hereditarily Lindelof nonseparable regular T2 space or the existence of a nonmetrizable normal Moore space. Neither of these problems has been satisfactorily answered yet. A less grandiose endeavour would be to construct an absolute example of a normal first countable ccc nonseparable space. As far as the author knows, no such absolute example exists in the literature to date. The example X* that we construct is also sigma compact and is a perfectpreimage of a metrizable" @default.
- W2081162393 created "2016-06-24" @default.
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- W2081162393 date "1979-01-01" @default.
- W2081162393 modified "2023-09-26" @default.
- W2081162393 title "A normal first countable ccc nonseparable space" @default.
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- W2081162393 doi "https://doi.org/10.1090/s0002-9939-1979-0521889-0" @default.
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