In this paper, we introduce a new closure operator termed the (δs)ω-closure operator and relate it to the δω-closure, θω-closure, and the usual closure operator In this paper, we introduce a new closure operator termed the (δs)ω-closure operator and relate it to the δω-closure, θω-closure, and the usual closure operator

This purpose of this paper is to introduce a new class of open sets namely semi-delta-open sets . Further, some basic topological concepts such as neighbourhood axioms, border point, exterior point, frontier point of a set are defined and their properties have been investigated. In addition, in terms of these open sets, semi-delta-closed functions and semi-delta-continuous functions are also defined and their properties have been discussed

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.

Al‐Jarrah et al. defined a new topological operator, namely, δ_ω‐closure operator, and proved that it lies between the δ‐closure operator and the usual closure operator. Al‐Ghour et al. defined θ_ω‐closure operator and discussed its properties. In this paper, it is proved that the δ_ω‐closure operator lies between the θ_ω‐closure operator and the usual closure operator. Also, sufficient conditions are given for the equivalence between the δ_ω‐closure operator and the θ_ω‐closure operator. Moreover, we define three new types of continuity, namely, δ_ω‐continuity, ω‐δ‐continuity, and almost δ_ω‐continuity, and discuss their properties. It is proved that the concepts of usual continuity and δ_ω‐continuity are independent of each other. In addition, the relationships between different types of continuity have been investigated. Further, some examples and counter examples are given.