The Josefson–Nissenzweig theorem and filters on $$\omega $$

For a free filter F on \(\omega \), endow the space \(N_F=\omega \cup \{p_F\}\), where \(p_F\not \in \omega \), with the topology in which every element of \(\omega \) is isolated whereas all open neighborhoods of \(p_F\) are of the form \(A\cup \{p_F\}\) for \(A\in F\). Spaces of the form \(N_F\) constitute the class of the simplest non-discrete Tychonoff spaces. The aim of this paper is to study them in the context of the celebrated Josefson–Nissenzweig theorem from Banach space theory. We prove, e.g., that, for a filter F, the space \(N_F\) carries a sequence \(\langle \mu _n:n\in \omega \rangle \) of normalized finitely supported signed measures such that \(\mu _n(f)\rightarrow 0\) for every bounded continuous real-valued function f on \(N_F\) if and only if \(F^*\le _K{\mathcal {Z}}\), that is, the dual ideal \(F^*\) is Katětov below the asymptotic density ideal \({\mathcal {Z}}\). Consequently, we get that if \(F^*\le _K{\mathcal {Z}}\), then: (1) if X is a Tychonoff space and \(N_F\) is homeomorphic to a subspace of X, then the space \(C_p^*(X)\) of bounded continuous real-valued functions on X contains a complemented copy of the space \(c_0\) endowed with the pointwise topology, (2) if K is a compact Hausdorff space and \(N_F\) is homeomorphic to a subspace of K, then the Banach space C(K) of continuous real-valued functions on K is not a Grothendieck space. The latter result generalizes the well-known fact stating that if a compact Hausdorff space K contains a non-trivial convergent sequence, then the space C(K) is not Grothendieck.