H-G-Spaces and H-Limit sets

Abstract

This paper develops a generalisation of the classical Cartan G-space within the framework of H-topology, a setting introduced by Fadhil to broaden the notion of openness in topological spaces. Building on H-open sets, H-continuous functions, H-irresolute maps, and H-compact spaces, we introduce H-thin sets and H-Cartan G-spaces. We further define two types of H-limit sets, denoted Λ^H(x) and J^H(x), for an arbitrary point x in a G-space, and we establish several characterisation theorems relating these sets to the orbit structure and the H-Cartan property. Concrete examples are provided throughout to illustrate each concept and to demonstrate that the inclusion Λ^H(x) , J^H(x) can be strict.