Selectively pseudocompact spaces

TL;DR

The paper investigates selective pseudocompactness variations when selections are allowed to be finite, scattered, discrete, or nowhere dense sets. It introduces the $\mathcal{A}$-SP framework and a natural hierarchy $1$-SP $\Rightarrow$ fin-SP $\Rightarrow$ ClScat-SP $\Rightarrow$ Scat-SP $\Rightarrow$ Nwd-SP $\Rightarrow$ pseudocompact, and demonstrates many of these implications are strict via explicit constructions in crowded spaces. A general separation method using base-enumerations and two families $\mathbb{G}(\mu,M)$ and $\mathbb{H}(\mu)$ derived from a monotone subadditive $\mu$ is developed to produce 0-dimensional $T_2$ spaces $X=\langle C\cup \omega_1,\tau\rangle$ with controlled accumulation behavior (e.g., $t(X)=\omega$ and $\,\omega_1$ left-separated), enabling separation of SP variants from pseudocompactness. The authors instantiate this framework to separate $1$-SP, fin-SP, ClScat-SP, Scat-SP, and Nwd-SP, and they prove a stronger result separating Scat-SP from Nwd-SP by constructing a crowded, 0-dimensional Hausdorff space of size $\omega_1$ in which all countable discrete subsets are closed and $Nwd_{\omega}(X)$-SP holds but $Scat_{\omega}(X)$-SP fails. These results clarify the strictness of the hierarchy among selective pseudocompactness notions and provide a versatile constructive method for tailoring accumulation properties in topological spaces.

Abstract

A novel selection principle was introduced by Dorantes-Aldama and Shakhmatov: a topological space $X$ is termed {\em selectively pseudocompact} if for any sequence $(U_n:n\in ω)$ of pairwise disjoint non-empty open sets of $X$, one can choose points $x_n\in U_n$ such that the sequence $(x_n:n\in ω)$ has an accumulation point. In this paper, we explore various versions of this principle when we permit the selection of finite, scattered, or nowhere dense sets instead of just singletons. We develop a method to prove that the aforementioned versions of selective pseudocompactness are indeed distinct from one another.

Theorems & Definitions (29)

• Definition 1.1
• Definition 1.2
• Lemma 1.3
• Proposition 1.4
• proof
• Theorem 1.5: CH
• Proposition 1.6
• proof
• Theorem 1.7: CH
• Definition 2.1