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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 ofthe simplest non-discrete Tychonoff spaces. The aim of this paper is to studythem in the context of the celebrated Josefson--Nissenzweig theorem from Banachspace theory. We prove, e.g., that, for a filter $F$, the space $N_F$ carries asequence $\langle\mu_n\colon n\in\omega\rangle$ of normalized finitelysupported signed measures such that $\mu_n(f)\to 0$ for every boundedcontinuous real-valued function $f$ on $N_F$ if and only if$F^*\le_K\mathcal{Z}$, that is, the dual ideal $F^*$ is Kat\v{e}tov below theasymptotic density ideal $\mathcal{Z}$. Consequently, we get that if$F^*\le_K\mathcal{Z}$, then: (1) if $X$ is a Tychonoff space and $N_F$ ishomeomorphic to a subspace of $X$, then the space $C_p^*(X)$ of boundedcontinuous real-valued functions on $X$ contains a complemented copy of thespace $c_0$ endowed with the pointwise topology, (2) if $K$ is a compactHausdorff space and $N_F$ is homeomorphic to a subspace of $K$, then the Banachspace $C(K)$ of continuous real-valued functions on $K$ is not a Grothendieckspace. The latter result generalizes the well-known fact stating that if acompact Hausdorff space $K$ contains a non-trivial convergent sequence, thenthe space $C(K)$ is not Grothendieck.

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