Core compactness and diagonality in spaces of open sets

TL;DR

This paper explores the conditions under which the space of open sets of a topological space, equipped with the Scott topology, exhibits properties like core compactness and diagonality, revealing deep connections with infraconsonance.

Contribution

It characterizes core compactness and diagonality of the open set space in terms of infraconsonance and Scott topology coincidences, providing new examples and clarifications.

Findings

Conditions for core compactness related to infraconsonance.

Characterization of diagonality without pretopology.

New examples illustrating relationships between topologicity, pretopologicity, and diagonality.

Abstract

We investigate when the space $\mathcal O_X$ of open subsets of a topological space $X$ endowed with the Scott topology is core compact. Such conditions turn out to be related to infraconsonance of $X$, which in turn is characterized in terms of coincidence of the Scott topology of $\mathcal O_X\times\mathcal O_X$ with the product of the Scott topologies of $\mathcal O_X$ at $(X,X)$. On the other hand, we characterize diagonality of $\mathcal O_X$ endowed with the Scott convergence and show that this space can be diagonal without being pretopological. New examples are provided to clarify the relationship between pretopologicity, topologicity and diagonality of this important convergence space.