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Higher stationarity and derived topologies on $\mathcal{P}_κ(A)$

M. Catalina Torres

TL;DR

This work extends Bagaria's ordinal theory of higher stationarity to the two-cardinal setting on $\\mathcal{P}_{\\kappa}({A})$ by introducing a sequence of topologies $\\langle \\tau_0, \\tau_1, abla \\rangle$ whose derived sets align with $n$-s-stationarity. It establishes that $1$-stationarity of $\\mathcal{P}_{\\kappa}({A})$ is equivalent to $\\kappa$ being weakly Mahlo, while higher levels require stronger large-cardinal hypotheses, mirroring the ordinal case. The paper constructs a robust topological framework in which the derived-topology approach generalises Bagaria's theorems (e.g., analogues of Theorems 2.11 and 3.1 from B2) to $\\mathcal{P}_{\\kappa}({A})$, and demonstrates that under total indescribability or $\\lambda$-supercompactness, extensive $n$-s-stationarity holds on $\\mathcal{P}_{\\kappa}({A})$ or $\\mathcal{P}_{\\kappa}(\\\lambda)$. It also develops isomorphisms preserving higher stationary notions to transfer results across two-cardinal settings, linking with strong-stationarity concepts and offering a framework for further exploration of reflection phenomena in two-cardinal contexts.

Abstract

Let $κ$ be a regular limit cardinal, $κ\subseteq A$. We study a notion of $n$-s-stationarity on $\mathcal{P}_κ(A)$. We construct a sequence of topologies $\langle τ_0, τ_1, \dots \rangle $ on $\mathcal{P}_κ(A)$ characterising the simultaneous reflection of a pair of $n$-s-stationary sets in terms of elements in the base of $τ_n$. This result constitutes a complete generalisation to the context of $\mathcal{P}_κ(A)$ of Bagaria's prior characterisation of $n$-simultaneous reflection in terms of derived topologies on ordinals.

Higher stationarity and derived topologies on $\mathcal{P}_κ(A)$

TL;DR

This work extends Bagaria's ordinal theory of higher stationarity to the two-cardinal setting on by introducing a sequence of topologies whose derived sets align with -s-stationarity. It establishes that -stationarity of is equivalent to being weakly Mahlo, while higher levels require stronger large-cardinal hypotheses, mirroring the ordinal case. The paper constructs a robust topological framework in which the derived-topology approach generalises Bagaria's theorems (e.g., analogues of Theorems 2.11 and 3.1 from B2) to , and demonstrates that under total indescribability or -supercompactness, extensive -s-stationarity holds on or . It also develops isomorphisms preserving higher stationary notions to transfer results across two-cardinal settings, linking with strong-stationarity concepts and offering a framework for further exploration of reflection phenomena in two-cardinal contexts.

Abstract

Let be a regular limit cardinal, . We study a notion of -s-stationarity on . We construct a sequence of topologies on characterising the simultaneous reflection of a pair of -s-stationary sets in terms of elements in the base of . This result constitutes a complete generalisation to the context of of Bagaria's prior characterisation of -simultaneous reflection in terms of derived topologies on ordinals.
Paper Structure (5 sections, 48 theorems, 40 equations)

This paper contains 5 sections, 48 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Notation and framework
  3. Higher stationary subsets of $\mathcal{P}_{\kappa}({A})$
  4. A topology for $\mathcal{P}_{\kappa}({A})$
  5. A sufficient condition for $n$-stationarity in $\mathcal{P}_{\kappa}({A})$

Key Result

Lemma 3.3

If $S \subseteq \mathcal{P}_{\kappa}({A})$ is $1$-stationary in $\mathcal{P}_{\kappa}({A})$, then $S$ is cofinal in $\mathcal{P}_{\kappa}({A})$.

Theorems & Definitions (96)

  • Definition 3.1: H. Brickhill, S. Fuchino and H. Sakai S1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • proof
  • Corollary 1
  • ...and 86 more