Calibers and point-finite cellularity of the space Cp(X) and some questions of S. Gul'ko and M. Hušek (Q1074903)

For any Tichonoff space X let Cp(X) denote the space of all continuous real-valued functions defined on X with the topology of pointwise convergence. A cardinal \(\tau\) is said to be a caliber of X in case every family with cardinality \(\tau\) of non-empty open subsets of X has a subfamily of cardinality \(\tau\) with a non-empty intersection. The aim of the present paper is to find conditions on a compact space X which will ensure that \(\aleph_ 1\) is a caliber of Cp(X). The authors show, among other things, that the following result is consistent with ZFC: a compact space X is metrizable if, and only if, \(\aleph_ 1\) is a caliber of Cp(X). They go on to show that a compact space X is metrizable if, and only if, \(\aleph_ 1\) and \(\aleph_ 2\) are both calibers of Cp(X), provided we assume that \(2^{\aleph_ 1}=\aleph_ 2\). For any set A let \(\Sigma_ A\) denote the set of real-valued functions f defined on A with the property that \(\{\) \(a\in A:\) \(| f(a)| \geq \epsilon \}\) is finite for every \(\epsilon >0\). Another of the author's results says that an extremally disconnected space X can be continuously and linearly injected into \(\Sigma_ A\) for some set A if, and only if, X is separable.