Hopf structures on closure spaces

Abstract

This study extends algebraic perspectives to non-topological closure spaces by introducing Hopf structures. We define closure Hopf spaces and groups, investigate their properties, and explore homotopy theory within this framework. Contravariant functors are established between the homotopy category of closure Hopf groups and the category of groups. We also introduce the concept of sub-CH groups for analyzing similar algebraic properties within CH group subsets. This research significantly advances our understanding of algebraic structures in closure spaces, and broadening the scope of mathematical exploration in this field.