New Continuity Concepts With Usual, Semi, and Semi‐ <i>ω</i> ‐Closure Operators

In this paper, we introduce new forms of continuity, namely, weakly θ s ‐continuity, weakly ‐continuity, almost θ s ‐continuity, and almost ‐continuity defined via closure operators. These concepts bridge the gap between classical and weak continuity and provide new insights into their relationships. We establish necessary and sufficient conditions under which these forms align with existing notions such as semi‐ θ s ‐continuity, ‐continuity, weak continuity, and usual continuity, especially under specific constraints on the domain or codomain space. Our findings highlight both the common ground and the key differences between weakly and almost continuity concepts. To support and illustrate our findings, the study incorporates a wide range of examples and counterexamples, providing a comprehensive understanding of these concepts.