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Hereditary \(\kappa\)-separability and the hereditary \(\kappa\)-Lindelöf property in function spaces. (English) Zbl 0666.54003
This paper is concerned with the smallest linear subspace \(L_ p(X)\) of \(C_ pC_ p(X)\) containing the Tychonoff space X. It is proved that \(L_ p(X)\) is hereditarily \(\kappa\)-Lindelöf (hereditarily \(\kappa\)- separable, resp.) if and only if \(X^{\omega}\) is hereditarily \(\kappa\)- Lindelöf (hereditarily \(\kappa\)-separable, resp.). Moreover, it is shown that a certain cardinal function of \(L_ p(X)\) called the weak pseudonet weight of \(L_ p(X)\) equals the net weight of X.
MSC:
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54C35 Function spaces in general topology
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