The basic of bipolar soft set theory stands for a mathematical instrument that brings together the soft set theory and bipolarity. Its definition is based on two soft sets, a set that provides positive information and other that gives negative. This paper mainly aims at defining a new bipolar soft generalized topological space; setting out of the point that the collection of bipolar soft sets forms the basis for the definition of the new concept is defined. Added to that, an investigation has been made of the four concepts of bipolar soft generalized, namely g-interior, g-closure, g-exterior and g-boundary. Furthermore, the main properties of bipolar soft generalized topological space (BSGT S) are established. This paper also attends to the discussion of the relations between these new definitions and the application of the given bipolar soft generalized topological spaces in a decision-making problem where an algorithm for this application has been suggested. Finally, to clarify and substantiate what the current work subsumes, some examples have been provided.

The aim of this paper is to give an interesting connection between two mathematical approaches to vagueness: binary bipolar soft sets and binary bipolar soft topology. The binary bipolar soft points are defined using binary bipolar soft sets. The binary bipolar soft set will be the binary bipolar soft union of its binary bipolar soft points. Moreover, the notion of binary bipolar soft topological spaces over two universal sets and a parameter set is proposed. Some topological properties of binary bipolar soft sets, such as binary bipolar soft open, binary bipolar soft closed, binary bipolar soft closure, binary bipolar soft interior, and binary bipolar soft boundary, are introduced. Some important properties of these classes of binary bipolar soft sets are investigated. Furthermore, the symmetry relation is compared between binary bipolar soft topology and binary soft topology on a common universe set. Finally, some results and counterexamples are demonstrated to explain this work.

In this paper, we originate a new class of open sets, namely $\mathtt{C}$-open sets, and we review its important properties. Then, some separation axioms of $\mathtt{C}$-open sets are introduced and investigated. In addition, we define the so-called $\mathtt{C}$-compact and ${\mathtt{C}}^{\prime }$-compact spaces via $\mathtt{C}$-open sets, and the theorems based on them are discussed with counterexamples. Moreover, we entitle the $\mathtt{C}$-continuous and ${\mathtt{C}}^{\prime }$-continuous functions by applying $\mathtt{C}$-open sets. In particular, several inferred properties of them and their connection with the other topological spaces are studied theoretically. Many examples are given to explain the concepts lucidly. The results established in this research paper are new in the field of topology.

In the present paper, we first introduced the notion of k-fuzzy $\gamma$-open (k-$\mathcal{F}$-$\gamma$-open) sets as a generalized novel class of fuzzy open ($\mathcal{F}$-open) sets on fuzzy topological spaces ($\mathcal{FTS}s$) in the sense of Šostak. The class of k-$\mathcal{F}$-$\gamma$-open sets is contained in the class of k-$\mathcal{F}$-$\beta$-open sets and contains all k-$\mathcal{F}$-semi-open and k-$\mathcal{F}$-pre-open sets. Also, we introduced the closure and interior operators with respect to the classes of k-$\mathcal{F}$-$\gamma$-closed and k-$\mathcal{F}$-$\gamma$-open sets and discussed some of their properties. After that, we defined and studied the notions of $\mathcal{F}$-$\gamma$-continuous (resp. $\mathcal{F}$-$\gamma$-irresolute) functions between $\mathcal{FTS}s$$(M,\Im )$ and $(N,\mathsf{Ϝ})$. However, we displayed and investigated the notions of $\mathcal{F}$-almost (resp. $\mathcal{F}$-weakly) $\gamma$-continuous functions, which are weaker forms of $\mathcal{F}$-$\gamma$-continuous functions. Next, we presented and characterized some new $\mathcal{F}$-functions via k-$\mathcal{F}$-$\gamma$-open and k-$\mathcal{F}$-$\gamma$-closed sets, called $\mathcal{F}$-$\gamma$-open (resp. $\mathcal{F}$-$\gamma$-irresolute open, $\mathcal{F}$-$\gamma$-closed, $\mathcal{F}$-$\gamma$-irresolute closed, and $\mathcal{F}$-$\gamma$-irresolute homeomorphism) functions. The relationships between these classes of functions were investigated with the help of some examples. We also introduced some new types of $\mathcal{F}$-separation axioms called k-$\mathcal{F}$-$\gamma$-regular (resp. k-$\mathcal{F}$-$\gamma$-normal) spaces via k-$\mathcal{F}$-$\gamma$-closed sets and discussed some properties of them. Lastly, we explored and studied some new types of $\mathcal{F}$-compactness called k-$\mathcal{F}$-almost (resp. k-$\mathcal{F}$-nearly) $\gamma$-compact sets.

In the present article, we define and investigate the notion of r-fuzzy γ-open (r-F-γ-open) sets as a generalized novel class of fuzzy open (F-open) sets on fuzzy topological spaces (F T Ss) in the sense of Šostak. This class is contained in the class of r-F-β-open sets and contains all r-F-pre-open and r-F-semi-open sets. However, we introduce the interior and closure operators with respect to the classes of r-F-γ-open and r-F-γ-closed sets, and study some of their properties. After that, we define and discuss the notions of F-γ-continuous (respectively (resp. for short) F-γ-irresolute) functions between F T Ss (M, ℑ) and (N, F). Also, we display and investigate the notions of F-almost (resp. F-weakly) γ-continuous functions, which are weaker forms of F-γ-continuous functions. We also showed that F-γ-continuity ⇒ F-almost γ-continuity ⇒ F-weakly γ-continuity, but the converse may not be true. Next, we present and characterize new F-functions via r-F-γ-open and r-F-γ-closed sets, called F-γ-open (resp. F-γ-irresolute open, F-γ-closed, F-γ-irresolute closed, and F-γ-irresolute homeomorphism) functions. The relationships between these classes of functions were discussed with the help of some examples. We also introduce some new types of F-separation axioms, called r-F-γ-regular (resp. r-F-γ-normal) spaces via r-F-γ-closed sets, and study some properties of them. Lastly, we explore and discuss some new types of F-compactness, called r-F-almost (resp. r-F-nearly) γ-compact sets using r-F-γ-open sets.

We develop foundational elements of a topology whose open sets are single-valued (T,I,N,F)-neutrosophic sets. The four coordinates represent truth, pure indeterminacy, neutrality, and falsehood, and the abbreviation TINF is used only for notational convenience. After fixing the order and complement needed to define inclusion, union, intersection, and duality, we introduce single-valued (T,I,N,F)-neutrosophic topological spaces and study closed sets, interior, closure, boundary, subspaces, products, and continuous maps. Classical topologies are represented naturally through crisp (T,I,N,F) characteristic sets, yielding an ordinary crisp core for every TINF topology. We also compare the resulting topological structure with related four-component theories: a coordinate permutation identifies it with quadripartitioned neutrosophic topology, while another identifies the underlying min--max open-set structure with the Turiyam value structure, although the published Fuzzy Neutrosophic Turiyam complement is not preserved. The main topological construction is the space of TINF points. We define the ordinary topology induced on this point space and prove that TINF continuity is equivalent to ordinary continuity of the induced point map. We then establish exact support-based correspondences for the separation axioms $T_0$, $T_1$, and $T_2$, and investigate connectedness and compactness. Although $V_{A\wedge B}=V_A\cap V_B$, the inclusion $\bigcup_\lambda V_{A_\lambda}\subseteq V_{\bigvee_\lambda A_\lambda}$ may be strict; this leads to strengthened notions of TINF connectedness and compactness that correspond exactly to ordinary connectedness and compactness of the induced point space.