Namespaces
Variants
Actions

Luzin hypothesis

From Encyclopedia of Mathematics
Revision as of 17:09, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

in set theory

The cardinality of the continuum is the cardinality of the set of all subsets of the countable ordinals, that is, . 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, . Martin's axiom (cf. Suslin hypothesis) and the negation of the continuum hypothesis together imply the Luzin hypothesis. The negation of the Luzin hypothesis, , 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.

References

[1] N.N. [N.N. Luzin] Lusin, "Sur les ensembles analytiques nuls" Fund. Math. , 25 (1935) pp. 109–131
[2] A. Mostowski, "Constructible sets ands applications" , North-Holland (1969)


Comments

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

References

[a1] K. Kunen, "Set theory" , North-Holland (1980)
How to Cite This Entry:
Luzin hypothesis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin_hypothesis&oldid=14782
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article