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- Title
The Josefson–Nissenzweig theorem and filters on ω.
- Authors
Marciszewski, Witold; Sobota, Damian
- Abstract
For a free filter F on ω , endow the space N F = ω ∪ { p F } , where p F ∉ ω , with the topology in which every element of ω is isolated whereas all open neighborhoods of p F are of the form A ∪ { p F } for A ∈ 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 ⟨ μ n : n ∈ ω ⟩ of normalized finitely supported signed measures such that μ n (f) → 0 for every bounded continuous real-valued function f on N F if and only if F ∗ ≤ K Z , that is, the dual ideal F ∗ is Katětov below the asymptotic density ideal Z . Consequently, we get that if F ∗ ≤ K 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.
- Subjects
BANACH spaces; HAUSDORFF spaces; CONTINUOUS functions; FUNCTION spaces; COMPACT spaces (Topology)
- Publication
Archive for Mathematical Logic, 2024, Vol 63, Issue 7/8, p773
- ISSN
0933-5846
- Publication type
Article
- DOI
10.1007/s00153-024-00920-x