Three problems in convergence theory

TL;DR

This paper investigates the complexity of classes of convergence structures, showing that paratopologies are simple, while hypotopologies are not under certain set-theoretic assumptions, and provides an example related to Hausdorff convergence.

Contribution

It establishes the simplicity of paratopologies, demonstrates the non-simplicity of hypotopologies under measurable cardinal assumptions, and offers a new example in Hausdorff convergence theory.

Findings

Paratopologies form a simple class.

Hypotopologies are not simple if measurable cardinals exist.

An example of subdiagonal convergence not being weakly diagonal.

Abstract

In this note it is proved that the class of paratopologies is simple and that under the assumption that the measurable cardinals form a proper class, the class of hypotopologies is not simple. Moreover, an example is given of a Hausdorff convergence with idempotent set adherence (subdiagonal convergence) that is not weakly diagonal.