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) |
Calibre. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Calibre&oldid=46189