Hereditarily Lindelöf spaces of continuous functions
For a Tychonoff space X, let \(C_ p(X)\) be the space of all continuous real functions on X in the topology of pointwise convergence and s(X) the supremum of cardinalities of discrete subspaces of X. Inequalities \(hd(X)\leq \aleph_ 0\) (resp. \(h\ell (X)\leq \aleph_ 0)\) denote that every subspace of X is separable (resp. Lindelöf). Further, SA means that each regular hereditarily separable space is hereditarily Lindelöf. The author showed the relationship between s, hd and \(h\ell\) for \(X^{\aleph_ 0}\), \(C_ p(X)\) and \((C_ p(X))^{\aleph_ 0}\) assuming SA, as below. Theorem (SA). If \(s(C_ p(X))\leq \aleph_ 0\), then \(s(X^{\aleph_ 0})=hd(X^{\aleph_ 0})=hl(X^{\aleph_ 0})\leq \aleph_ 0\) and \(s((C_ p(X))^{\aleph_ 0})=hd((C_ p(X))^{\aleph_ 0})\leq \aleph_ 0\) hold. - The following is a partial solution of Velichko's problem in the case of \(\aleph_ 0:\) Corollary (SA). If \(s(C_ p(X))\leq \aleph_ 0\), then \(s((C_ p(X))^{\aleph_ 0})=s(C_ p(X))\) and \(h\ell ((C_ p(X))^{\aleph_ 0})=h\ell (C_ p(X))\).
scientific article; zbMATH DE number 4175959