Topology and category for singular product spaces

TL;DR

This paper explores the topology and category theory of higher Baire and Cantor spaces for singular cardinals, focusing on their structure and the properties of meagre subsets.

Contribution

It investigates the topology of function spaces for singular cardinals and analyzes the cardinal characteristics of meagre sets within these spaces.

Findings

Analyzed the structure of higher Baire and Cantor spaces for singular cardinals.

Studied the properties of the ideal of $$-meagre subsets in these spaces.

Abstract

For $\kappa$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^\kappa\kappa$ and ${}^\kappa2$ (endowed with the ${<}\kappa$-box topology) have been relatively well-studied, but less is known about the case where $\kappa$ is singular. We will consider several spaces of functions and box topologies that could serve as higher Baire and Cantor spaces for singular cardinals. The ultimate focus of the article lies in studying cardinal characteristics of the ideal of $\kappa$-meagre subsets of these spaces.