Ordinary Smooth Topological Spaces

Abstract

In this paper, we introduce the concept of ordinary smooth topology on a set X by considering the gradation of openness of ordinary subsets of X. And we obtain the result [Corollary 2.13] : An ordinary smooth topology is fully determined its decomposition in classical topologies. Also we introduce the notion of ordinary smooth [resp. strong and weak] continuity and study some its properties. Also we introduce the concepts of a base and a subbase in an ordinary smooth topological space and study their properties. Finally, we investigate some properties of an ordinary smooth subspace.Key words : ordinary smooth (co)topological space, r-level and strong r-level, ordinary smooth [resp. weak and strong] continuity, ordinary smooth open [resp. closed] mapping, ordinary smooth subspace, ordinary smooth base [resp. subbase].

Closures and Interiors Redefined, and Some Types of Compactness in Ordinary Smooth Topological Spaces

“…Definition 2.4 [9] Let X be a nonempty set. Then a mapping C : 2 X → I is called an ordinary smooth cotopology (in short, osct) on X or a gradation of closedness of ordinary subsets of X if C satisfies the following axioms :…”

“…Definition 2.7 [9] Let (X, τ ) be an osts and let r ∈ I. Then we define two ordinary subsets of X as follows :…”

“…Recently, Hur et al [9] gave the similar definition called an ordinary smooth topolgy on X as a mapping τ : 2 X → I satisfying three axioms, where 2 denotes the two points set {0, 1}. In particular, Chae et al [10] studied the set OST(X) of all ordinary smooth topologies on X in the sense of a lattice.…”

The Lattice of Ordinary Smooth Topologies

“…They all have investigated the gradation of openness[resp. closedness] of fuzzy sets in a set X. Lim et al [5] introduce the notion of ordinary smooth topologies by considering the gradation of openness[resp. closedness] of ordinary subsets of X.…”

“…Lim et al [5] introduce the notion of ordinary smooth topologies by considering the gradation of openness[resp. closedness] of ordinary subsets of X.…”