A new class between theta open sets and theta omega open sets

Area of Science:

• General Topology
• Set Theory

Background:

• The study of topological spaces involves various closure operators and separation axioms.
• Existing operators like theta-closure and eta-closure have specific properties within topology.

Purpose of the Study:

• To introduce and investigate a novel topological operator, the omega-closure operator.
• To define and analyze new concepts such as omega-open sets and omega-regularity.
• To explore the relationships between these new concepts and existing topological structures.

Main Methods:

• Definition of the omega-closure operator, positioned between theta-closure and eta-closure.
• Introduction of omega-open sets derived from the omega-closure operator.
• Characterization of topological spaces using the omega-closure operator.
• Definition of omega-regularity as a new separation axiom.
• Investigation of continuity concepts including omega-continuity and its variants.

Main Results:

• Established relationships between the omega-closure operator and other closure operators (theta-closure, eta-closure, usual closure).
• Introduced a new topology based on omega-open sets.
• Provided mapping theorems related to the new topology.
• Characterized omega-topological spaces via the omega-closure operator.
• Defined omega-regularity, a separation axiom strictly between omega-regularity and regularity.
• Demonstrated the equivalence of omega-regularity to a specific condition in topological spaces.
• Introduced and studied omega-continuity, omega-theta-continuity, weak omega-continuity, and faint omega-continuity.

Conclusions:

• The omega-closure operator provides a new framework for studying topological spaces.
• The newly defined omega-open sets and omega-regularity offer novel perspectives in general topology.
• The research contributes to the hierarchy of separation axioms and continuity types in topological spaces.