Notes on linearly H-closed spaces and od-selection principles

TL;DR

This paper explores the properties and examples of linearly H-closed spaces, a class between feeble compactness and H-closedness, and examines related selection principles and set-theoretic implications.

Contribution

It introduces the concept of linearly H-closed spaces, provides examples across various classes, and investigates their properties under different set-theoretic assumptions.

Findings

Existence of linearly H-closed spaces that are collectionwise normal and Fréchet-Urysohn.

PFA implies no first countable normal linearly H-closed non-compact spaces under certain conditions.

Ostaszewski space is an example of a perfectly normal linearly H-closed space built with $\

Abstract

A space is called linearly H-closed iff any chain cover possesses a dense member. This property lies strictly between feeble compactness and H-closedness. While regular H-closed spaces are compact, there are linearly H-closed spaces which are even collectionwise normal and Fr\'echet-Urysohn. We give examples in other classes, and ask whether there is a first countable normal linearly H-closed non-compact space in ZFC. We show that PFA implies a negative answer if the space is moreover either locally separable or locally compact and locally ccc. Ostaszewski space (built with $\diamondsuit$) is an example which is even perfectly normal. We also investigate Menger-like properties for the class of od-covers, that is, covers whose members are open and dense.