Soft Continuous Mappings in Soft Closure Spaces

Soft closure spaces are a new structure that was introduced very recently. These new spaces are based on the notion of soft closure operators. This work aims to provide applications of soft closure operators. We introduce the concept of soft continuous mappings and soft closed (resp. open) mappings, support them with examples, and investigate some of their properties.


1.Introduction
The concept of soft sets was first introduced by Molodtsov [1] in 1999 as a general mathematical tool for dealing with uncertain objects. Soft set theory has been applied in many directions, e.g., stability and regularization [1], game theory and operations research [1], soft analysis [1], group theory [2], general topology [3], etc. Moreover, in the structure of closure spaces, Gowri and Jegadeesan [4] and Krishnaveni and Sekar [5] introduced and studied ̆e ch soft closure spaces. In the classical soft set theory, because of the fuzzy existence of the parameters, a condition can be complicated in the real world. In this respect, classical ̆e ch soft closure spaces were expanded to ̆e ch soft closure spaces [6,7,8]. Recently, Ekram and Majeed introduced the notion of soft closure spaces [9] as an expansion to this concept in the ordinary case of the set theory that was introduced by ̌e ch [10].
Continuity is an important notion in general topology, soft topology, and closure spaces as well as all branches of mathematics and quantum physics. Kharal and Ahmad [11] presented the concept of a mapping on the classes of soft sets that is a central notion for the advancement of every new field of mathematical science. An idea of soft mapping was presented and some of its properties were studied ISSN: 0067-2904 Ekram and Majeed Iraqi Journal of Science, 2021, Vol. 62, No. 8, pp: 2676-2684 7722 in [12]. Boonpok [13] defined and studied the concept of continuity in closure spaces. Our work in the present paper is dedicated to presenting the concept of soft continuity in soft closure spaces. In Section 3, we introduce the concept of soft continuous and study some of their properties. Also, the notion of soft closed (resp. open) mappings is introduced. In Section 4, several properties and characterizations related to soft projection mappings, closed (resp. open) soft sets, and soft continuous (resp. soft closed) mappings in the product of soft closure spaces are discussed.

2.Preliminaries
In this section, we introduce the basic definitions and results of soft set theory and soft closure spaces that will be needed in the sequel. Definition 2.1 [1]. A soft set over the universe set is defined by a mapping . Then, can be represented by the set ( ) . We denote the family of all soft sets over by . Definition 2.2 [14]. A null soft set, denoted by ̃ , is a soft set over such that for all , (empty set). Definition 2.3 [14]. An absolute soft set, denoted by ̃, is a soft set over such that for all , . Definition 2.4 [15]. Let and be two soft sets over .
Definition 2.11 [17]. Let and . The Cartesian product is defined by where , for all . From this definition, the soft set is a soft set over and its universe parameter is . The pairs of projections , and , determine, respectively, the morphisms from to and from to , where and [18].
Definition 2.12 [9]. An operator ̃ is called a soft closure operator (soft-, for short) on , if for all the following axioms are satisfied: and is a closed soft set. Definition 2.13 [9]. Let ( ̃ be a soft-cs and let . Let ̃ defined by ̃ ̃ ̃ . Then, ̃ is called the relative soft closure operator on induced by ̃. The triple ( ̃ is called a soft closure subspace (soft-c.subsp, for short) of ( ̃ . Theorem 2.14 [9]. Let ̃ be a family of soft-cs's. If ∏ is an open soft set in the product soft- is an open soft set in ̃ for all .

3.Soft continuous mappings
In this section, we introduce the concept of soft continuous (resp. soft closed) mappings between soft closure spaces, with some examples to explain these notions. Also, some properties related to these concepts are given.

Soft continuous mappings between product soft closure spaces
In this section, we study some properties of soft continuous mappings in the product soft closure space. First, we show that the soft projection map is soft closed and continuous.

Conclusions
Soft closure spaces are a very new concept and an important topic for investigators because it is more general as compared to the concept of soft topological spaces. The notions of soft continuous and soft closed (resp. open) mappings were introduced in this paper, and some related properties and theorems were developed. To explain our notions, we put forward some examples.