Calibre

of a topological space

A cardinal number such that every family of cardinality , consisting of non-empty open subsets of a topological space , contains a subset , also of cardinality , with non-empty intersection, i.e. . A regular uncountable cardinal number is a calibre of a topological product , , if and only if is a calibre of every factor . 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 , then satisfies the Suslin condition. In some models of set theory the converse is almost true, namely, Martin's axiom and the condition imply the following: If a space satisfies the Suslin condition, then every uncountable family of non-empty open sets in contains an uncountable centred subfamily. In particular, in this model, the cardinal number 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 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 is a caliber of if and only if for every collection of non-empty open subsets of there is a set of size such that .

One also considers precalibers: a cardinal number is a precaliber of if and only if for every collection of non-empty subsets of there is a set of size such that has the finite intersection property (i.e. the intersection of any finite number of 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 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)