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 ![]() ![]() |
[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