On θ continuity

Area of Science:

• Topology
• Real Analysis
• Set Theory

Background:

• Continuity is a fundamental concept in topology.
• Existing continuity definitions have limitations in certain applications.
• Closure operators provide a framework for defining topological properties.

Purpose of the Study:

• Introduce and define four novel classes of continuous functions: θ-continuous, ω-θ-continuous, weakly θ-continuous, and faintly θ-continuous.
• Investigate the theoretical properties of these new function classes.
• Establish relationships between these new classes and existing ones.

Main Methods:

• Utilizing the closure operator and the θω-closure operator.
• Developing new definitions based on these operators.
• Employing theoretical mathematical proofs and constructions.

Main Results:

• Successfully defined four new classes of continuous functions.
• Derived key properties and characteristics for each new class.
• Identified interrelationships and distinctions among the proposed function types.

Conclusions:

• The introduction of these new function classes expands the landscape of continuity in topology.
• The study provides a foundational understanding of their behavior and potential applications.
• Further research can explore specific uses and implications of these novel continuous functions.