Calibre

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.

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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.

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