Hereditary \(\kappa\)-separability and the hereditary \(\kappa\)-Lindelöf property in function spaces. (English) Zbl 0666.54003

Johnson, Roy A.; Wajch, Eliza; Wilczyński, Wladyslaw

Commentat. Math. Univ. Carol. 30, No. 1, 75-80 (1989).

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

Keywords:

pointwise convergence; hereditary \(\kappa\)-separability; hereditary \(\kappa\)-Lindelöf property; weak pseudonet weight