Remarks on Pre-ℐ-Regular Pre-ℐ-Open Sets

Area of Science:

• Topology
• Set Theory

Background:

• Introduction of pre-I-open sets as a fundamental concept.
• Exploration of existing literature on generalized open sets in topological spaces.

Purpose of the Study:

• To define and characterize a new class of sets: pre-I-regular pre-I-open sets.
• To investigate the fundamental properties of these newly defined sets.
• To introduce and examine related concepts: extremally pre-I-disconnectedness, locally pre-I-indiscreetness, and pre-I-regular sets.

Main Methods:

• Utilizing the definitions of pre-I-open and pre-I-closed sets.
• Characterization of pre-I-regular pre-I-open sets through their properties.
• Derivation of equivalence relations based on the introduced concepts.

Main Results:

• Successful characterization of pre-I-regular pre-I-open sets.
• Identification and study of fundamental properties of these sets.
• Establishment of equivalence relations involving pre-I-regular pre-I-open sets and related notions.

Conclusions:

• The study successfully introduces and characterizes pre-I-regular pre-I-open sets.
• New topological properties and relations have been established.
• The findings contribute to the understanding of generalized open sets in topology.