For a free filter $F$ on $\omega$, let $N_F=\omega\cup\{ p_F\}$, where $p_F\not\in\omega$, be equipped with the following topology: every element of $\omega$ is isolated whereas all open neighborhoods of $p_F$ are of the form $A\cup\{ p_F\}$ for $A\in F$. The aim of this paper is to study spaces of the form $N_F$ in the context of the Nikodym property of Boolean algebras. By $\mathcal{ AN}$ we denote the class of all those ideals $\mathcal{ I}$ on $\omega$ such that for the dual filter $\mathcal{ I}^*$ the space $N_{ \mathcal{ I}^*}$ carries a sequence $\langle\mu_n\colon n\in\omega\rangle$ of finitely supported signed measures such that $\|\mu_n\|\rightarrow\infty$ and $\mu_n(A)\rightarrow 0$ for every clopen subset $A\subseteq N_{ \mathcal{ I}^*}$. We prove that $\mathcal{ I}\in\mathcal{ AN}$ if and only if there exists a density submeasure $\varphi$ on $\omega$ such that $\varphi(\omega)=\infty$ and $\mathcal{ I}$ is contained in the exhaustive ideal $\mbox{ Exh}(\varphi)$. Consequently, we get that if $\mathcal{ I}\subseteq\mbox{ Exh}(\varphi)$ for some density submeasure $\varphi$ on $\omega$ such that $\varphi(\omega)=\infty$ and $N_{ \mathcal{ I}^*}$ is homeomorphic to a subspace of the Stone space $St(\mathcal{ A})$ of a given Boolean algebra $\mathcal{ A}$, then $\mathcal{ A}$ does not have the Nikodym property. We observe that each $\mathcal{ I}\in\mathcal{ AN}$ is Kat\v{ e}tov below the asymptotic density zero ideal $\mathcal{ Z}$, and prove that the class $\mathcal{ AN}$ has a subset of size $\mathfrak{ d}$ which is dominating with respect to the Kat\v{ e}tov order $\leq_K$, but $\mathcal{ AN}$ has no $\leq_K$-maximal element. We show that for a density ideal $\mathcal{ I}$ it holds $\mathcal{ I}\not\in\mathcal{ AN}$ if and only if $\mathcal{ I}$ is totally bounded if and only if the Boolean algebra $\mathcal{ P}(\omega)/\mathcal{ I}$ contains a countable splitting family.
Categories: math.LO math.FA math.GN