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.

References

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

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.
 [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 $\sigma$-ideals on $\omega_1$" 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)