Topological complexity of ideal limit points

2024/07/16 by Marek Balcerzak, Balcerzak, Marek, Szymon Gła̧b +3 · 1 citation
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #Commutative Algebra and Its Applications #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Graph theory and applications #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2407.12160

openalex publication_date 2024/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an ideal I on the nonnegative integers ω and a Polish space X, let \mathscrL(I) be the family of subsets S⊆ X such that S is the set of I-limit points of some sequence taking values in X. First, we show that \mathscrL(I) may attain arbitrarily large Borel complexity. Second, we prove that if I is a G_δσ-ideal then all elements of \mathscrL(I) are closed. Third, we show that if I is a simply coanalytic ideal and X is first countable, then every element of \mathscrL(I) is simply analytic. Lastly, we studied certain structural properties and the topological complexity of minimal ideals I for which \mathscrL(I) contains a given set.

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