Soft super-continuity and soft delta-closed graphs.

Introducing a strong form of soft continuity between soft topological spaces is significant because it can contribute to our growing understanding of soft topological spaces and their features, provide a basis for creating new mathematical tools and methods, and have significant applications in various fields. In this paper, we define soft super-continuity as a new form of soft mapping. We present various characterizations of this soft concept. Also, we show that soft super-continuity lies strictly between soft continuity and soft complete continuity and that soft super-continuity is a strong form of soft δ-continuity. In addition, we give some sufficient conditions for the equivalence between soft super-continuity and other related concepts. Moreover, we characterize soft semi-regularity in terms of super-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and products by soft super-continuity. In addition to these, we study the relationship between soft super-continuity and soft δ-continuity with their analogous notions in general topology. Finally, we give several sufficient conditions on a soft mapping to have a soft δ-closed graph.

Soft C-continuity and soft almost C-continuity between soft topological spaces.

We present the notions of soft c-continuity and soft nearly c-continuity, which are weaker versions of soft continuity and soft almost continuity, respectively. We obtain several characterizations for these two concepts. We show that soft c-continuity and soft almost c-continuity are preserved under soft restrictions. In addition, we investigate the conditions under which the composition of two soft c-continuous and soft almost c-continuous functions is soft c-continuous and soft almost c-continuous. Moreover, via soft N-open sets, we give a characterization of the soft compactness of soft topological spaces over finite sets of parameters. In addition, we show that for a given soft topological space (L,Ω,F), the collection of soft regular open sets with soft compact complements forms a soft base for some coarser soft topology. We demonstrate that (L,Ω⁎,F) are all soft compacts. Furthermore, we demonstrate that Ω=Ω⁎ if (L,Ω,F) or (L,Ω⁎,F) is soft compact and soft Hausdorff. Finally, we investigate the correspondences between the novel notions in soft topology and their general topological analogs.

On soft submaximal spaces.

In the present paper, detailed properties of soft submaximal topological spaces have been discussed. It starts by examining how well soft submaximal spaces behave when subjected to some operations, such as: taking soft subspaces, soft products, soft topological sum and images/preimages under certain soft mappings. Furthermore, several characterizations of soft submaximal spaces are obtained with respect to various types of soft subsets of soft topological spaces and their simple extensions.

Soft ω-regular open sets and soft nearly Lindelöfness.

In this paper, we define the notion of "soft ω-regular openness," which lies exactly between "soft regular openness" and "soft ω-openness." Through soft ω-regular open sets, we introduce the notions of soft semi ω-regularity as a weaker form of soft semi regularity and soft almost ω-regularity as a strong form of soft almost regularity. We prove that soft ω-regular open sets of a soft topological space form a soft topology. Also, we prove that soft semi ω-regularity and soft almost ω-regularity are independent notions. In addition, we reveal the relationships between soft topology and classical (parametric) topology. Finally, we characterize soft nearly Lindelöfness and improve several results related to soft nearly Lindelöfness using the concept of soft ω-regular open sets.

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

We define θ N -closure operator as a new topological operator which lies between the θ-closure and the θ ω -closure. Some relationships between this new operator and each of θ-closure, θ ω -closure, and usual closure are obtained. Via θ N -closure operator, we introduce θ N -open sets as a new topology. Some mapping theorems related to the new topology are given. T 2 topological spaces are characterized in terms of θ N -closure operator. Also, we use N -open sets to define N -regularity as a new separation axiom which lies strictly between ω-regularity and regularity. For a given topological space ( Y , σ ) , we show that N -regularity is equivalent to the condition σ = σ θ N . Finally, θ N -continuity, N -θ-continuity, weak θ N -continuity, and faint θ N -continuity are introduced and studied.

On θ ω continuity.

We use the closure and the theta omega closure operators to introduce θ ω -continuous, ω-θ-continuous, weakly θ ω -continuous and faintly θ ω -continuous as new four classes of functions. We obtain several properties, relationships, examples and counter-examples related to them.

On some generated soft topological spaces and soft homogeneity.

We introduce soft homogeneity as an extension of homogeneity in ordinary topological spaces. Based on the generated soft topology of a given indexed family of classical topologies inspite of a one topology given by Terepeta in [16], we investigate soft minimal open set and homogeneity relation between the generated soft topology and the given indexed family of topologies. We introduce several results, examples and counterexamples.