The existence of continuous weak selections and orderability-type properties in products and filter spaces

TL;DR

This paper investigates the conditions under which filter spaces and their products possess continuous weak selections and orderability properties, establishing that certain product conditions imply hereditary paracompactness of suborderable images.

Contribution

It demonstrates that a closed continuous image of a suborderable space is hereditarily paracompact if its product with a non-discrete space admits a separately continuous weak selection.

Findings

A closed continuous image of a suborderable space is hereditarily paracompact under certain product conditions.

Product spaces with a non-discrete factor influence the orderability properties of filter spaces.

Conditions for continuous weak selections in products of filter spaces are characterized.

Abstract

Orderability, weak orderability and the existence of continuous weak selections on filter spaces (i.e., spaces with a single non-isolated point) and their products are discussed. We prove that a closed continuous image X of a suborderable space must be hereditarily paracompact provided that its product X\times Y with some non-discrete space Y has a separately continuous weak selection.