Weakly web-compact Banach spaces \(C(X)\), and \(Lip_0(M)\), \(\mathcal{F}(M)\) over metric spaces \(M\) (Q6542375)

A space, \(X\), is said to be web-compact (see [\textit{J. Orihuela}, J. Lond. Math. Soc., II. Ser. 36, No. 1--2, 143--152 (1987; Zbl 0608.46007)]) if there are a subset~\(\Sigma\) of~\(\omega^\omega\) and a family \(\{A_\alpha:\alpha\in\Sigma\}\) of subsets of~\(X\) such that \(\bigcup_{\alpha\in\Sigma}A_\alpha\) is dense in~\(X\), and if \(\alpha\in\Sigma\) then when \(x_n\in C_{\alpha\mathbin\upharpoonright n}\) for all~\(n\) then the sequence~\((x_n)_n\) has a cluster point in~\(X\). Here \(C_{\alpha\upharpoonright n}\) denotes \(\bigcup\{A_\beta:\beta\in\Sigma\) and \(\beta\mathbin\upharpoonright n = \alpha\mathbin\upharpoonright n\}\).\par The author investigates this notion in the context of Banach spaces with the weak topology and in particular spaces~\(C_p(X)\). The results are too numerous to summarize here comprehensively; a small sample may suffice. A Banach space is web-compact in the weak topology iff it is a Lindelöf \(\Sigma\)-space. If \(X\)~is compact and infinite then it is Gul'ko-compact iff \(C(X)_w\) is web-compact iff \(C_p(X)\) has a web-compact subset with a dense span.