On Qp-Closed Sets in Topological Spaces

Abstract

In the present paper, we will propose the novel notions (e.g., Qp-closed set, Qp-open set, Qp-continuous mapping, Qp-open mapping, and Qp-closed mapping) in topological spaces. Then, we will discuss the basic properties of the above notions in detail. The category of all Qp-closed (resp. Qp-open) sets is strictly between the class of all preclosed (resp. preopen) sets and gp-closed (resp. gp-open) sets. Also, the category of all Qp-continuity (resp. Qp-open (Qp-closed) mappings) is strictly among the class of all precontinuity (resp., preopen (preclosed) mappings) and gp-continuity (resp. gp-open (gp-closed) mappings). Furthermore, we will present the notions of Qp-closure of a set and Qp-interior of a set and explain some of their fundamental basic properties. Several relations are equivalent between two different topological spaces. The novel two separation axioms (i.e., Qp-ℝ0 and Qp-ℝ1) based on the notion of Qp-open set and Qp-closure are investigated. The space of Qp-ℝ0 (resp., Qp-ℝ1) is strictly between the spaces of pre-ℝ0 (resp., pre-ℝ1) and gp-ℝo (resp., gp-ℝ1). Finally, some relations and properties of Qp-ℝ0 and Qp-ℝ1 spaces are explained.