Closed Subsets of Noetherian Generalized Topological Spaces

Eman Almuhur¹^*, Husam Miqdad², Manal Al-labadi³, Mohammad I. Idrisi⁴

¹Department of Mathematics, Applied Science Private University, Amman, Jordan

²Department of Basic Science / Scientific, Applied Science Private University, Amman, Jordan

³Department of Mathematics, University of Petra, Amman, Jordan

⁴Department of Mathematics, Chandigarh University, Punjab, India

Emails: e_almuhur@asu.edu.jo; hmiqdad@hotmail.com; manal.allabadi@uop.edu.jo; mhdimranidrisi@gmail.com

Abstract

In the final years of the 20th century, the notion of generalized topological spaces was introduced, marking a significant shift in the field of topology. This paper focuses on a subset of  on a non-empty set  that is closed under arbitrary unions, defining a generalized topology and subsequently a generalized topological space (GTS) denoted by . Within this framework, we explore the concept of Noetherian generalized topological spaces and delve into the properties of closed subsets within the Noetherian GTS. The investigation reveals that subspaces of a Noetherian GTS , with the induced topology, inherit the Noetherian property and exhibit finitely many non-empty irreducible components. Furthermore, the study extends to the analysis of hereditary properties, regular , , irreducible closed subsets, and the product properties of closed subsets under continuous functions. We also establish the closure property of finite unions in Noetherian GTS and clarify the homeomorphic nature of Noetherian GTS   to itself.

Keywords: GTS, Noetherian, continuous function; fuzzy topology; neutrosophic topology.