Luzin hypothesis

in set theory

The cardinality of the continuum is the cardinality of the set of all subsets of the countable ordinals, that is, $2^{\aleph_0}=2^{\aleph_1}$. Luzin's hypothesis is compatible with the Zermelo–Fraenkel system of axioms of set theory and the axiom of choice. N.N. Luzin [1] considered this hypothesis as an alternative to the continuum hypothesis, that is, $2^{\aleph_0}=\aleph_1<2^{\aleph_1}$. Martin's axiom (cf. Suslin hypothesis) and the negation of the continuum hypothesis together imply the Luzin hypothesis. The negation of the Luzin hypothesis, $2^{\aleph_0}<2^{\aleph_1}$, is also sometimes called the Luzin hypothesis. The Luzin hypothesis, denoted by (HL), or its negation, which is denoted by (LH), are used in the proof of a number of theorems in general topology. For example, (LH) is equivalent to one of the following assertions: any compact space of cardinality not exceeding the cardinality of the continuum has an everywhere-dense subspace that satisfies the first axiom of countability; any dyadic compact Hausdorff space of cardinality not exceeding the cardinality of the continuum is metrizable. The following propositions follow from (LH): any normal space that satisfies the first axiom of countability and the Suslin condition is collection-wise normal; any separable normal Moore space is metrizable.

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For the consistency of Luzin's hypothesis see also [a1].

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