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Weak covering properties of weak topologies. (English) Zbl 0886.54014
In 1980, S. Gul’ko posed the problem whether the space \(C_p(K)\) is hereditarily meta-Lindelöf for every compact space \(K\). Later R. Hanswell asked whether such a space is weakly \(\theta\)-refinable. It is shown in the paper that the space \(C_p(\beta \omega_1)\) is a counterexample to both questions, and \(C_p (\beta \omega_1)\) is not even weakly \(\delta \theta\)-refinable. Furthermore, it is consistent with ZFC that there exists a scattered compact space \(K\) such that \(C_p(K)\) is not weakly \(\delta \theta\)-refinable. On the other hand, the authors prove that for every Banach space \(B\) of density at most \(\aleph_1\), the weak topology of \(B\) is hereditarily meta-Lindelöf and similarly, \(C_p(K)\) has this property for every compact space \(K\) with \(w(K) \leq\aleph_1\). Several interesting results about the spaces of the form \(C_p (\overline T)\) are obtained where \(\overline T\) is the one-point compactification of a tree \(T\) with the usual tree topology.

54C35 Function spaces in general topology
46B20 Geometry and structure of normed linear spaces
54E20 Stratifiable spaces, cosmic spaces, etc.
54D30 Compactness
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