The Josefson-Nissenzweig theorem and filters on

For a free filter F on $$ , endow the space $$ , where $$ , with the topology in which every element of $$ is isolated whereas all open neighborhoods of $$ are of the form $$ for $$ . Spaces of the form $$ 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 $$ carries a sequence $$ of normalized finitely supported signed measures such that $$ for every bounded continuous real-valued function f on $$ if and only if $$ , that is, the dual ideal $$ is Katětov below the asymptotic density ideal $$ . Consequently, we get that if $$ , then: (1) if X is a Tychonoff space and $$ is homeomorphic to a subspace of X, then the space $$ of bounded continuous real-valued functions on X contains a complemented copy of the space $$ endowed with the pointwise topology, (2) if K is a compact Hausdorff space and $$ 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.