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Calibre

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of a topological space $ X $

A cardinal number $ \tau $ such that every family $ \mathfrak B $ of cardinality $ \tau $, consisting of non-empty open subsets of a topological space $ X $, contains a subset $ \mathfrak B ^ \prime \subset \mathfrak B $, also of cardinality $ \tau $, with non-empty intersection, i.e. $ \cap \{ {U } : {U \in \mathfrak B ^ \prime } \} \neq \emptyset $. A regular uncountable cardinal number $ \tau $ is a calibre of a topological product $ \prod X _ \alpha $, $ \alpha \in A $, if and only if $ \tau $ is a calibre of every factor $ X _ \alpha $. The property of being a calibre is preserved under continuous mappings; every uncountable regular cardinal number is a calibre of any dyadic compactum. If the first uncountable cardinal number is a calibre of a space $ X $, then $ X $ satisfies the Suslin condition. In some models of set theory the converse is almost true, namely, Martin's axiom and the condition $ \aleph _ {1} < 2 ^ {\aleph _ {0} } $ imply the following: If a space $ X $ satisfies the Suslin condition, then every uncountable family of non-empty open sets in $ X $ contains an uncountable centred subfamily. In particular, in this model, the cardinal number $ \aleph _ {1} $ is a calibre for every compactum with the Suslin condition. In some other models of set theory, a compactum with the Suslin condition exists for which $ \aleph _ {1} $ is not a calibre.

References

[1] N.A. Suslin, "On the product of topological spaces" Trudy. Mat. Inst. Steklov , 24 (1948) (In Russian)

Comments

The spelling caliber is more common.

Usually, calibers are defined using indexed collections of open sets. In that case a cardinal number $ \kappa $ is a caliber of $ X $ if and only if for every collection $ \{ {U _ \alpha } : {\alpha \in \kappa } \} $ of non-empty open subsets of $ X $ there is a set $ A \subset \kappa $ of size $ \kappa $ such that $ \cap _ {\alpha \in A } U _ \alpha \neq \emptyset $.

One also considers precalibers: a cardinal number $ \kappa $ is a precaliber of $ X $ if and only if for every collection $ \{ {U _ \alpha } : {\alpha \in \kappa } \} $ of non-empty subsets of $ X $ there is a set $ A \subset \kappa $ of size $ \kappa $ such that $ \{ {U _ \alpha } : {\alpha \in \kappa } \} $ has the finite intersection property (i.e. the intersection of any finite number of $ U _ \alpha $ is non-empty). Thus, Martin's axiom (cf. Suslin hypothesis) plus the negation of the continuum hypothesis imply that every space satisfying the Suslin condition has $ \aleph _ {1} $ as a precaliber, while for a compact space its calibers and precalibers are the same.

References

[a1] S. Argyros, A. Tsarpalias, "Calibers of compact spaces" Trans. Amer. Math. Soc. , 270 (1982) pp. 149–162
[a2] S. Broverman, J. Ginsburg, K. Kunen, F.D. Tall, "Topologies determined by -ideals on " Canad. J. Math. , 30 (1978) pp. 1306–1312
[a3] W.W. Comfort, S. Negrepontis, "Chain conditions in topology" , Cambridge Univ. Press (1982)
[a4] I. Juhász, "Cardinal functions. Ten years later" , MC Tracts , 123 , Math. Centre (1980)
How to Cite This Entry:
Calibre. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Calibre&oldid=46189
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article