Continuum Without Non-Block Points

TL;DR

This paper characterizes the structure of certain continua in the Stone-Čech remainder of the half-plane, establishing equivalences involving dense semicontinua, ultrafilters, and Q-points, and explores implications for the continuum's block points.

Contribution

It provides new equivalences relating dense semicontinua, ultrafilter orderings, and Q-points in the context of the continuum without non-block points, addressing a question posed by the author.

Findings

Equivalence between dense semicontinua and ultrafilter properties.

NCF is characterized by the absence of proper dense semicontinua and non-block points.

Indecomposable continua contain a maximum semicontinuum with dense interior.

Abstract

For any composant $E \subset \mathbb H^*$ and corresponding near-coherence class $\mathscr E \subset \omega^*$ we prove the following are equivalent : (1) $E$ properly contains a dense semicontinuum. (2) Each countable subset of $E$ is contained in a dense proper semicontinuum of $E$. (3) Each countable subset of $E$ is disjoint from some dense proper semicontinuum of $E$. (4) $\mathscr E $ has a minimal element in the finite-to-one monotone order of ultrafilters. (5) $\mathscr E $ has a $Q$-point. A consequence is that NCF is equivalent to $\mathbb H^*$ containing no proper dense semicontinuum and no non-block points. This gives an axiom-contingent answer to a question of the author. Thus every known continuum has either a proper dense semicontinuum at every point or at no points. We examine the structure of indecomposable continua for which this fails, and deduce they contain a maximum semicontinuum with dense interior.