The purpose of this paper is to introduce a new structure called primal. Primal is the dual structure of grill. Like ideal, the dual of filter, this new structure also generates a new topology named primal topology. We introduce a new operator using primal, which satisfies Kuratowski closure axioms. Mainly, we prove that primal topology is finer than the topology of a primal topological space. Also, we provide the structure of the base of primal topology and prove other fundamental results related to this new structure. Furthermore, we not only discuss some of this new structure’s properties but also enrich it with many examples.

Recently, Acharjee, Özkoc and Issaka introduced a new structure in topology named primal. Primal is the dual structure of grill. The main purpose of this paper is to introduce an operator using primal and obtain some of its fundamental properties. Also, we define the notion of topology suitable for a primal. We not only obtain some characterizations of this new notion, but also investigate many properties.

Proximity space is one of the common topics in mathematics, computer science, and pattern recognition. Recently, Acharjee et al. introduced a new structure named primal in mathematics. Thus, the main purpose of this paper is to introduce and study primal-proximity spaces. Also, we define two new operators via primal proximity spaces and investigate some of their fundamental properties. In addition, we obtain a new topology, which is weaker than old one, via these new operators. Moreover, we not only discuss some of their properties but also enrich with some examples.

The main purpose of this paper is to introduce and study two new operators $ (\cdot)_R^{\diamond} $ and $ cl_R^{\diamond}(\cdot) $ via primal, which is a new notion. We show that the operator $ cl_R^{\diamond}(\cdot) $ is a Kuratowski closure operator, while the operator $ (\cdot)_R^{\diamond} $ is not. In addition, we prove that the topology on $ X $, shown as $ \tau_R^{\diamond}, $ obtained by means of the operator $ cl_R^{\diamond}(\cdot), $ is finer than $ \tau_{\delta}, $ where $ \tau_{\delta} $ is the family of $ \delta $-open subsets of a space $ (X, \tau). $ Moreover, we not only obtain a base for the topology $ \tau_R^{\diamond} $ but also prove many fundamental results concerning this new structure. Furthermore, we provide many counterexamples related to our results.

This paper aims to introduce and study two new operators $(.)^{\diamond}_{\omega}$ and $cl^{\diamond}_{\omega}(\cdot)$ by utilizing the notion of primal defined by S. Acharjee et al. Also, we investigate some fundamental properties of them. In addition, we showed that the operator $cl^{\diamond}_{\omega}(.)$ satisfied the Kuratowski closure axioms. Therefore, we obtain a new topology denoted by $\tau^{\diamond}_{\omega},$ which is finer than the original one. Moreover, the topology $\tau^{\diamond}_{\omega}$ obtained via the operator $cl^{\diamond}_{\omega}(\cdot)$ is finer than $\tau_{\omega}$, where $\tau_{\omega}$ is the family of all $\omega$-open subsets of a primal topological space $(X,\tau,\mathcal{P}).$ Furthermore, we not only examine the fundamental properties of this class of sets but also provide some counterexamples.

The main purpose of this work is to study the ideal topology defined by the minimal and maximal ideals on a topological space. Also, we define and investigate the concepts of ideal quotient and annihilator of any subfamily of 2 X , where 2 X is the power set of X . We obtain some of their fundamental properties. In addition, several relationships among the above notions have been discussed. Moreover, we define a new topology on an ideal topological space, called sharp topology, via the sharp operator defined in this study, which turns out to be finer than the original topology. Furthermore, a decomposition of open sets (in the original topology) has been obtained. Finally, we conclude our work with some interesting applications.