The class of web-compact spaces (in sense of Orihuela), which encompasses a number of spaces, like Lindelöf \(\Sigma \) -spaces (called also countably determined), Quasi-Suslin spaces, separable spaces, etc., applies to distinguish a class of weakly web-compact Banach spaces E whose dual unit ball is weak \(^{*}\) -sequentially compact, consequently Banach spaces without quotients isomorphic to \(\ell _{\infty }.\) We prove however that for a Banach space E the space \(E_w\) (i.e. E with the weak topology) is web-compact if and only if \(E_w\) is a Lindelöf \(\Sigma \) -space if and only if \(E_w\) contains a web-compact total subset. Consequently, for compact X the space \(C(X)_w\) is web-compact if and only if X is Gul’ko compact if and only if \(C(X)_w\) is a Lindelöf \(\Sigma \) -space if and only if \(C_p(X)\) contains a web-compact total subset. If X is compact and \(C(X)_w\) is web-compact, then \(C_p(X)\) contains a complemented copy of the space \((c_{0})_p=\{(x_{n})\in \mathbb R^{\omega }: x_{n}\rightarrow 0\}\) with the topology of \(\mathbb {R}^{\omega }\) but does not admit quotients isomorphic to \((\ell _{\infty })_{p}=\{(x_{n})\in \mathbb R^{\omega }: \sup _n|x_{n}|<\infty \}\) . We characterize weakly web-compact Banach spaces \(Lip_0(M)\) of Lipschitz functions on metric spaces M and their predual \(\mathcal {F}(M)\) . In fact, \(Lip_0(M)_w\) is web-compact if and only if M is separable. Illustrating examples are provided.
