Soft Closure Spaces

In this paper, the concept of soft closure spaces is defined and studied its basic properties. We show that the concept soft closure spaces are a generalization to the concept of Čech soft closure spaces introduced by Krishnaveni and Sekar. In addition, the concepts of subspaces and product spaces are extended to soft closure spaces and discussed some of their properties.


Introduction
There are many mathematical tools obtainable for dealing with an imperfect knowledge or for modelling complex systems such as probability theory, fuzzy set theory, rough set theory and also in computer science, engineering, physics, social sciences, economics, and medical sciences, etc. All these tools require the pre-specification of some parameters to start with. To conquer these obstacles, in 1999 Molodtsov [12] proposed a new mathematical tool, namely soft set theory to model uncertainty, which associates a set with a set of parameters. After Molodtsove's activity work, in 2003 Maji et al. [10] presented and studied several basic notions of soft set theory and some operation between two soft sets. The Applications of the theory of soft sets have been in many areas of mathematics. In 2011, Shabir and Naz [14] defined and studied the soft topological space. In 2014, El-Sheikh and Abd El-Latif [5] initiated the notion of supra soft topological spaces, which is wider and more general than the class of soft topological spaces.
The concept of closure space were introduced by ̌e ch [3] in 1968, where is a mapping defined on the power set of a set satisfying: and , the mapping called closure operator on . A closure operator is called ̌e ch closure operator, if satisfies: and then is called ̌e ch closure space. ̌e ch closure spaces studied by several authors and in several directions. In 1985, Mashhour and Ghanim [11] introduced the concept of ̌e ch closure spaces in fuzzy setting. Independently, in 2014, Gowri and Jegadeesan [7] and Krishnaveni and Sekar [8] defined and studied ̆e ch closure spaces in soft setting. Recently, Majeed [9] using fuzzy soft sets to define the concept of ̆e ch fuzzy soft closure spaces.
In this work, motivated by the theory of soft sets we introduced the notion of soft closure spaces. In Section 3, the concept of soft closure spaces is defined. Also, the notion of closed (respectively, open) soft sets in soft closure spaces is defined and give the basic properties of them with several examples to explain these concepts. In addition, we show our notion of soft closure space in more general than the notion of ̆e ch soft closure spaces that defined by Krishnaveni and Sekar [8] (see Proposition 3.4). Moreover, we find for every soft closure space there exists a supra soft topology associative with it (see Remark 3.18). In Section 4, the soft closure subspace of a soft closure space is defined and studied with details. We discuss the relationships between the closed (respectively, open) soft sets in the soft-cs and its soft-c.subsp (see Proposition 4.7 and Theorems 4.10 and 4.12) Finally, Section 5 is devoted to introduce the notion of the product of soft closure spaces and studied its basic properties.

Preliminaries
In this section we recall some basic definitions and results of soft set theory defined and discussed by various authors. Throughout this paper, refers to the initial universe, denote the power set of and is the set of all parameters for .  [13] Let be a supra soft topological space and be a non-empty subset of . Then, is called the supra soft relative topology on and is called a supra soft subspace of .

The Basic Structures of Soft Closure Spaces
This section is devoted to introduce the notion of soft closure spaces and discussed the basic properties of these spaces.

Definition 3.1
An operator ̃ is called a soft closure operator (soft-, for short) on , if for all the following axioms are satisfied:

The triple
̃ is called a soft closure space (soft-, for short).
Next, we give two examples to explain the notion in Definition 3.1.

Example 3.2 Let and
. Define a soft-co ̃ as follows: Proposition 3.14 The intersection of any collection of closed soft sets in a soft-cs is a closed soft set.
Then, it is easy to verify that ̃ and ̃ are soft-co's on and ̃ is finer than ̃ since for all , ̃ ̃ .

Soft closure subspaces
In this section we introduce the notion of soft closure subspace of a soft-cs and investigate some properties of its. The following Theorem give the condition to be the converse of Proposition 4.7 is hold in general.

Theorem 4.10
Let ( ̃ be a soft-cs and ( , ̃ be a closed soft subspace of ( ̃ . If is a closed soft set of ( ̃ then is a closed soft set of ( ̃ Proof: To prove is a closed soft set of ( ̃ we must show ̃ = .

Remark 4.11
In Theorem 4.10, the soft set ̃ is a closed soft set in is a necessary condition for this theorem. We can explain that in more details.

Remark 4.13
The convers of Theorem 4.12 is not true as the following example shows. , where ∏ and ∏ denotes to the Cartesian product of the sets and , , respectively as follows: Then, the operator ̃ is a soft closure operator on ∏ .
Proof: We must prove ̃ satisfies the axioms -of Definition 3.1.