Vol 8, No 3 (2017)

Available online throug

ISSN 2229 – 5046

A STUDY ON SOFT PRE

-

OPEN SETS

V. E. SASIKALA*, D. SIVARAJ

Meenakshi Academy of Higher Education and Research,

Meenakshi University, Chennai, Tamil Nadu, India.

(Received On: 01-03-17; Revised & Accepted On: 27-03-17)

ABSTRACT

T

he objective of this paper is to describe the concepts of soft pre-open sets and soft pre-closed sets in soft topological spaces by studying the basics of soft set theory by D. Molodstov’s description.

Keywords: soft set, soft topology, soft open sets and soft closed sets.

Subject Classification: 54A10, 54A05.

1. INTRODUCTION

Several researchers followed Molodtsov [4] 1999, after his introduction of Soft set theory as a common mathematical application in dealing with the vagueness of not well defined objects, many researchers followed him. When several proved mathematical applications have more clarity while applying on formal modeling, reasoning and computing, but few of the engineering sciences, life sciences, social sciences and ecological sciences does not have clarity many times. Shabir and Naz [10] described the soft topological spaces and its basic notations in detail.

The equality of two soft sets, subset and superset of Soft set, complement of a soft set, null soft set and absolute soft set with examples were well defined by Maji [7]. As the researchers now have several latest sophisticated techniques, they have now applied them in operations research, Riemans integration, Game theory, theory of probability and researched several basic notations of soft set theory. Naim et al., [8] presented the foundations of the theory of soft topological spaces to open the door for experiments for soft mathematical concepts and structures that are based on soft set-theoretic operations. Recently Soft topology has been studied in depth in the papers [1, 2, 3, 5, 6, 9, 11, 12, 13, 14]. In this paper, we make a theoretical study of the new set called soft pre-open set and soft pre-closed sets in soft topological space.

2. PRELIMINARIES

The following definitions are essential for the development of the paper.

Definition 2.1 [4]: Let 𝑈𝑈 be an initial universe and 𝐸𝐸 be a set of parameters. Let 𝑃𝑃(𝑈𝑈) denote the power set of 𝑈𝑈. The pair (𝐹𝐹,𝐸𝐸) or simply 𝐹𝐹_{𝐸𝐸 ,} is called a soft set over 𝑈𝑈, where 𝐹𝐹 is a mapping given by 𝐹𝐹:𝐸𝐸 → 𝑃𝑃(𝑈𝑈). In other words, a soft set over 𝑈𝑈 is a parameterized family of subsets of the universe. For 𝑒𝑒 ∈ 𝑈𝑈, 𝐹𝐹(𝑒𝑒) may be considered as the set of e-approximate elements of the soft set 𝐹𝐹. The collection of all soft sets over 𝑈𝑈and 𝐸𝐸 is denoted by 𝑆𝑆(𝑈𝑈). If 𝐴𝐴⊆𝐸𝐸, then the pair (𝐹𝐹,𝐴𝐴) or simply 𝐹𝐹_𝐴𝐴, is called a soft set over 𝑈𝑈, where 𝐹𝐹 is a mapping 𝐹𝐹:𝐴𝐴 → 𝑃𝑃(𝑈𝑈). Note that for 𝑒𝑒∉𝐴𝐴,𝐹𝐹(𝑒𝑒) =∅.

Definition 2.2 [11]: The union of two soft sets of 𝐹𝐹_𝐵𝐵 and 𝐺𝐺_𝐶𝐶 over the common universe 𝑈𝑈 is the soft set 𝐻𝐻𝐷𝐷, where 𝐵𝐵 and 𝐶𝐶are subsets of the parameter set 𝐸𝐸, 𝐷𝐷=𝐵𝐵 ∪ 𝐶𝐶 and for all e∈D,𝐻𝐻(𝑒𝑒) =𝐹𝐹(𝑒𝑒) if e∈B−C,

𝐻𝐻(𝑒𝑒) =𝐺𝐺(𝑒𝑒) if e∈C−B and 𝐻𝐻(𝑒𝑒) =𝐹𝐹(𝑒𝑒)∪ 𝐺𝐺(𝑒𝑒) if e∈B∩C, we write 𝐹𝐹_𝐵𝐵Ũ𝐺𝐺_𝐶𝐶 =𝐻𝐻_𝐷𝐷.

Definition 2.3 [11]: The intersection of two soft sets of 𝐹𝐹_𝐵𝐵 and 𝐺𝐺_𝐶𝐶 over the common universe 𝑈𝑈 is the soft set 𝐻𝐻𝐷𝐷, where D=B∩C and for all e∈D, 𝐻𝐻(𝑒𝑒) =𝐹𝐹(𝑒𝑒)∩ 𝐺𝐺(𝑒𝑒) if D = B∩C. We write 𝐹𝐹𝐵𝐵∩� 𝐺𝐺𝐶𝐶=𝐻𝐻𝐷𝐷.

Corresponding Author: V. E. Sasikala*

1

Definition 2.4 [11]: Let 𝐹𝐹_𝐵𝐵 and 𝐺𝐺_𝐶𝐶 be soft sets over a common universe set 𝑈𝑈 and 𝐵𝐵,𝐶𝐶⊆𝐸𝐸. Then 𝐹𝐹_𝐵𝐵 is a soft subset of 𝐺𝐺_𝐶𝐶, denoted by 𝐹𝐹_𝐵𝐵⊆�𝐺𝐺_𝐶𝐶, if (i) B⊆C and (ii) for all e∈B, 𝐹𝐹(𝑒𝑒) =𝐺𝐺(𝑒𝑒). Also 𝐺𝐺_𝐶𝐶, is called the soft super set of 𝐹𝐹_𝐵𝐵 and is denoted by 𝐹𝐹_𝐵𝐵⊇�𝐺𝐺_𝐶𝐶.

Definition 2.5 [11]: The soft sets 𝐹𝐹_𝐵𝐵 and 𝐺𝐺_𝐶𝐶over a common universe set 𝑈𝑈 are said to be soft equal, if 𝐹𝐹_𝐵𝐵⊆�𝐺𝐺_𝐶𝐶, and 𝐹𝐹_𝐵𝐵⊇�𝐺𝐺_𝐶𝐶. Then we write 𝐹𝐹_𝐵𝐵=𝐺𝐺_𝐶𝐶.

Definition 2.6 [11]: A soft set 𝐹𝐹_𝐵𝐵over 𝑈𝑈 is called a null soft set denoted by 𝐹𝐹_𝜙𝜙, if for all e∈B, F (e) = ϕ.

Definition 2.7 [10]: The relative complement of a soft set 𝐹𝐹_𝐴𝐴, denoted by 𝐹𝐹_𝐴𝐴𝑐𝑐, is defined by the approximate function _𝑓𝑓

𝐴𝐴𝑐𝑐(𝑒𝑒) =𝑓𝑓𝐴𝐴𝑐𝑐(𝑒𝑒), where 𝑓𝑓𝐴𝐴𝑐𝑐(𝑒𝑒) is the complement of the set 𝑓𝑓𝐴𝐴(𝑒𝑒), that is 𝑓𝑓𝐴𝐴𝑐𝑐(𝑒𝑒) =𝑈𝑈 − 𝑓𝑓𝐴𝐴(𝑒𝑒) for all e∈E.

It is easy to see that (𝐹𝐹_𝐴𝐴𝑐𝑐)𝑐𝑐 =𝐹𝐹_𝐴𝐴, 𝐹𝐹_ϕ𝑐𝑐=𝐹𝐹_𝐸𝐸 and 𝐹𝐹_E𝑐𝑐 =𝐹𝐹_ϕ.

Definition 2.8 [11]: Let 𝑈𝑈 be an initial universe and 𝐸𝐸 be a set of parameters. If 𝐵𝐵⊆𝐸𝐸, the soft set 𝐹𝐹_𝐵𝐵 over 𝑈𝑈

is called an absolute soft set, if for all e∈B, 𝐹𝐹(𝑒𝑒) =𝑈𝑈.

The following is the definition of soft topology used by various authors.

Definition 2.9 [1]: Let 𝑈𝑈 be an initial universe and 𝐸𝐸 be a set of parameters. Let 𝜏𝜏 be a subcollection of 𝑆𝑆(𝑈𝑈),

the collection of soft sets defined on 𝑈𝑈. Then 𝜏𝜏 is a soft topology if it satisfies the following conditions. (i) 𝐹𝐹_ϕ ,𝐹𝐹_𝐸𝐸 ∈ 𝜏𝜏

(ii) The union of any number of soft sets in 𝜏𝜏 belongs to 𝜏𝜏. (iii)The intersection of any two soft sets in 𝜏𝜏 belongs to 𝜏𝜏.

If this definition is considered for further development, then the results are similar to that of results in topological spaces. Therefore, throughout the paper the following definition of soft topology is used.

Definition 2.10 [10]: Let 𝑈𝑈 be an initial universe and𝐸𝐸 be a set of parameters. Let 𝐹𝐹_𝐴𝐴∈ 𝑆𝑆(𝑈𝑈). A soft topology

on 𝐹𝐹_𝐴𝐴, denoted by 𝜏𝜏̃, is a collection of soft subsets of 𝐹𝐹_𝐴𝐴 having the following properties: (i) 𝐹𝐹_ϕ , 𝐹𝐹_𝐴𝐴 ∈ 𝜏𝜏̃

(ii) �𝐹𝐹_{𝐴𝐴𝑖𝑖}⊆�𝐹𝐹_𝐴𝐴∶ 𝑖𝑖 ∈ 𝐼𝐼�⊆�𝜏𝜏̃⇒Ũ_𝑖𝑖∈I𝐹𝐹_{𝐴𝐴𝑖𝑖} ∈ 𝜏𝜏̃ (iii)�𝐹𝐹_𝐴𝐴

𝑖𝑖⊆�𝐹𝐹𝐴𝐴∶1≤ 𝑖𝑖 ≤ 𝑛𝑛 , 𝑛𝑛 ∈ Ν�⊆�𝜏𝜏̃⇒ ∩�𝑖𝑖=1𝑛𝑛 𝐹𝐹𝐴𝐴𝑖𝑖 ∈ 𝜏𝜏̃. (iv)The pair (𝐹𝐹_𝐴𝐴 ,𝜏𝜏̃) is called a soft topological spaces.

Definition 2.11 [9]: Let (𝐹𝐹_𝐴𝐴 ,𝜏𝜏̃) be a soft topological space in 𝐹𝐹_𝐴𝐴. Elements of 𝜏𝜏̃ are called soft open sets. A soft set 𝐹𝐹_𝐵𝐵 in 𝐹𝐹_𝐴𝐴 is said to be a softclosed set in 𝐹𝐹_𝐴𝐴, if its relative complement 𝐹𝐹_𝐵𝐵𝑐𝑐, belongs to 𝜏𝜏̃.

Definition 2.12 [12]: Let (𝐹𝐹_𝐴𝐴 ,𝜏𝜏̃) be a soft topological space and 𝐹𝐹_𝐵𝐵be a soft subset of 𝐹𝐹_{𝐴𝐴 .} (i) The soft interior of 𝐹𝐹_𝐵𝐵 is the soft set int(𝐹𝐹_𝐵𝐵) = Ũ{𝐹𝐹_𝐶𝐶: 𝐹𝐹_𝐶𝐶 is soft open and 𝐹𝐹_𝐶𝐶⊆�𝐹𝐹_𝐵𝐵}. (ii) The soft closure of 𝐹𝐹_𝐵𝐵 is the soft set cl(𝐹𝐹_𝐵𝐵) =∩� {𝐹𝐹_𝐶𝐶: 𝐹𝐹_𝐶𝐶 is soft closed and 𝐹𝐹_𝐵𝐵⊆�𝐹𝐹_𝐶𝐶}.

Its clearly int(𝐹𝐹_𝐵𝐵) is the largest soft open set contained in 𝐹𝐹_𝐵𝐵 and cl(𝐹𝐹_𝐵𝐵) is the smallest soft closed set containing 𝐹𝐹_𝐵𝐵.

Lemma 2.13 [12]: Let (𝐹𝐹_𝐴𝐴 ,𝜏𝜏̃) be a soft topological space and 𝐹𝐹_𝐵𝐵 and 𝐹𝐹_𝐶𝐶 be a soft subsets of 𝐹𝐹_𝐴𝐴. Then the following holds.

(i) int(int(𝐹𝐹_𝐵𝐵)) = int(𝐹𝐹_𝐵𝐵).

(ii) F_B ⊆� F_C implies int(F_{B )}⊆� int(F_C).

(iii)int(𝐹𝐹_𝐵𝐵)∩� int(𝐹𝐹_𝐶𝐶) = int(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶). (iv)int(𝐹𝐹_{𝐵𝐵 )} Ũ int(𝐹𝐹_{𝐶𝐶 )} ⊆� int(𝐹𝐹_𝐵𝐵Ũ 𝐹𝐹_{𝐶𝐶 ).}

(v) 𝐹𝐹_𝐵𝐵is soft open set if and only if 𝐹𝐹_𝐵𝐵= int(𝐹𝐹_𝐵𝐵).

Lemma 2.14 [12]: Let (𝐹𝐹_𝐴𝐴 ,𝜏𝜏̃) be a soft topological space and 𝐹𝐹_𝐵𝐵and 𝐹𝐹_𝐶𝐶 be a soft set in 𝐹𝐹_𝐴𝐴. Then the following hold.

(i) cl(cl(𝐹𝐹_𝐵𝐵)) = cl(𝐹𝐹_𝐵𝐵).

(ii) 𝐹𝐹_𝐵𝐵 ⊆� 𝐹𝐹_𝐶𝐶 implies cl(𝐹𝐹_𝐵𝐵)⊆� cl(𝐹𝐹_𝐶𝐶).

(iii)cl(𝐹𝐹_𝐵𝐵)∩� cl(𝐹𝐹_𝐶𝐶)⊇� cl(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶). (iv) cl(𝐹𝐹_{𝐵𝐵 )}Ũcl(𝐹𝐹_{𝐶𝐶 ) = cl(}𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶).

(v) 𝐹𝐹_𝐵𝐵 is soft closed set if and only if 𝐹𝐹_𝐵𝐵= cl(𝐹𝐹_𝐵𝐵).

Proposition 2.16 [2]: Let (𝐹𝐹_𝐴𝐴 ,τ�) be a soft topological space over 𝐹𝐹_𝐴𝐴. Then the following hold. (i) 𝐹𝐹_∅,𝐹𝐹_𝐴𝐴 are soft closed sets in 𝐹𝐹_𝐴𝐴.

(ii) The union of any two soft closed sets is a soft closed set in 𝐹𝐹_𝐴𝐴. (iii)The intersection of any two soft closed sets is a soft closed set in 𝐹𝐹_𝐴𝐴.

Proposition 2.17 [12]: If �𝐹𝐹_{𝐵𝐵𝛼𝛼}�𝛼𝛼 ∈ 𝐼𝐼� is a collection of soft sets, then the following hold. (i) _Ũint (𝐹𝐹_{𝐵𝐵𝛼𝛼})⊆�int (Ũ𝐹𝐹_{𝐵𝐵𝛼𝛼})

(ii) _Ũ cl(𝐹𝐹_{𝐵𝐵𝛼𝛼})⊆�cl(Ũ𝐹𝐹_{𝐵𝐵𝛼𝛼}).

Lemma 2.18 [13]:

(i) For every soft open set 𝐹𝐹_𝐵𝐵 in a soft topological space (𝐹𝐹_𝐴𝐴 ,τ�) and every soft set 𝐹𝐹_𝐶𝐶, we have cl(𝐹𝐹_𝐶𝐶) ∩� 𝐹𝐹_𝐵𝐵⊆� cl(𝐹𝐹_𝐶𝐶∩� 𝐹𝐹_𝐵𝐵).

(ii) For every soft closed set in 𝐹𝐹_𝐵𝐵 a soft topological space (𝐹𝐹_𝐴𝐴 ,τ�) and every soft set 𝐹𝐹_𝐶𝐶, we have int(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶)⊆� int(𝐹𝐹_𝐵𝐵)Ũ𝐹𝐹_𝐶𝐶.

Proposition 2.19 [7]: If 𝐹𝐹_{𝐵𝐵 and} 𝐹𝐹_𝐶𝐶 are any two soft sets in (𝐹𝐹_𝐴𝐴 ,τ�), then the following hold 𝐹𝐹𝐴𝐴−(𝐹𝐹𝐵𝐵− 𝐹𝐹𝐶𝐶) = (𝐹𝐹𝐴𝐴− 𝐹𝐹𝐵𝐵)Ũ(𝐹𝐹𝐴𝐴− 𝐹𝐹𝐶𝐶).

3. SOFT PRE-OPEN SET AND SOFT PRE-CLOSED SET

This section is devoted to the study of soft pre-open sets and soft pre- closed sets.

Definition 3.1: Let (𝐹𝐹_𝐴𝐴 ,𝜏𝜏̃) be a soft topological space and 𝐹𝐹_𝐵𝐵 be a soft subset of 𝐹𝐹_𝐴𝐴. 𝐹𝐹_𝐵𝐵 is said to be a soft pre-open set, if 𝐹𝐹_𝐵𝐵⊆� int(cl( F_B)).

Example 3.2: Let 𝑈𝑈={a, b, c}, 𝐸𝐸={e₁, e₂, e₃}, 𝐴𝐴={e₁, e₂}⊆�𝐸𝐸.

𝐹𝐹𝐴𝐴={(e1, {a, b}) , (e2, {b, c})}, 𝐹𝐹1 ={(e1, {a})}, 𝐹𝐹2 = {(e1, {b})} 𝐹𝐹3={(e1, {a, b})},

𝐹𝐹4 ={(e2, {b})} ,𝐹𝐹₅={(e1, {c})},𝐹𝐹₆ ={(e2, {b, c})},𝐹𝐹₇={(e1 , {a}) , (e2, {b})},

𝐹𝐹8={(e1 , {a}) , (e2, {c})} ,𝐹𝐹₉={(e1 , {a}) (e2 , {b, c})},𝐹𝐹₁₀={(e1 , {b}) (e2, {b})},𝐹𝐹₁₁={(e1, {b}) (e2, {c})},

𝐹𝐹12={(e1, {b}) (e2, {b, c})} , 𝐹𝐹₁₃={(e1, {a, b}) , (e2, {b})}, 𝐹𝐹14={(e1, {a, b}) , (e2, {c})}, 𝐹𝐹₁₅= 𝐹𝐹_𝐴𝐴 and𝐹𝐹₁₆= 𝐹𝐹_∅.

Let τ�= {𝐹𝐹_∅, 𝐹𝐹_𝐴𝐴 ,𝐹𝐹₁,𝐹𝐹₃,𝐹𝐹₇,𝐹𝐹₉,𝐹𝐹₁₃}. Then (𝐹𝐹_𝐴𝐴 ,𝜏𝜏̃) is a soft topological space. The family of all soft closed sets is {𝐹𝐹∅, 𝐹𝐹𝐴𝐴 ,𝐹𝐹12,𝐹𝐹6,𝐹𝐹11,𝐹𝐹2,𝐹𝐹5 }. The family of soft pre-open sets is {𝐹𝐹𝐴𝐴, 𝐹𝐹∅ ,𝐹𝐹1,𝐹𝐹3,𝐹𝐹4,𝐹𝐹7,𝐹𝐹8,𝐹𝐹9,𝐹𝐹10,𝐹𝐹12,𝐹𝐹13,𝐹𝐹14}. The

family of soft pre-closed sets is _{𝐹𝐹

𝐴𝐴, 𝐹𝐹∅ ,𝐹𝐹1,𝐹𝐹2,𝐹𝐹4,𝐹𝐹5,𝐹𝐹6,𝐹𝐹8,𝐹𝐹10,𝐹𝐹11,𝐹𝐹12,𝐹𝐹14}.

Theorem 3.3: Every soft open set in a soft topological space (𝐹𝐹_𝐴𝐴 ,𝜏𝜏̃) is a soft pre-open set.

Proof: The proof follows from the Definition 3.1.

The following Example 3.4 shows that the converse implication of Theorem 3.3 is not true.

Example 3.4: Consider the soft topological space of Example 3.2. Here 𝐹𝐹₄, 𝐹𝐹₈, 𝐹𝐹₁₀, 𝐹𝐹_{12 and} 𝐹𝐹_{14 are soft}

pre-open sets but not soft pre-open sets, since 𝐹𝐹₄, 𝐹𝐹₈, 𝐹𝐹₁₀, 𝐹𝐹₁₂, 𝐹𝐹₁₄∉τ�.

Remark 3.5: 𝐹𝐹_∅ and 𝐹𝐹_𝐴𝐴 are always soft pre-closed sets and soft pre-open sets.

Theorem 3.6: Arbitrary union of soft pre-open sets is a soft pre-open set.

Remark 3.7: Arbitrary intersection of soft pre-closed sets is a soft pre-closed set.

The following Example 3.8 shows that the finite intersection of soft pre-open sets need not be a soft pre-open set.

Theorem 3.9: If 𝐹𝐹_𝐵𝐵is a soft pre-open set such that 𝐹𝐹_𝐶𝐶⊆�𝐹𝐹_𝐵𝐵⊆�cl(𝐹𝐹_𝐶𝐶), then 𝐹𝐹_𝐶𝐶 is also a soft pre-open set.

Proof: 𝐹𝐹_𝐵𝐵⊆�cl(𝐹𝐹_𝐶𝐶) implies that cl(𝐹𝐹_𝐵𝐵)⊆�cl(𝐹𝐹_𝐶𝐶) and so 𝐹𝐹_𝐵𝐵⊆�int(cl(𝐹𝐹_𝐵𝐵))⊆� int(cl(𝐹𝐹_𝐶𝐶)) which implies that 𝐹𝐹𝐶𝐶⊆� int(cl(𝐹𝐹𝐶𝐶)). Therefore, 𝐹𝐹𝐶𝐶 is a soft pre-open set.

Theorem 3.10: Let 𝐹𝐹_𝐵𝐵be a soft subset of a soft topological space (𝐹𝐹_𝐴𝐴 ,τ�). Then 𝐹𝐹_𝐵𝐵is soft pre-closed set if and only if cl(int(𝐹𝐹_𝐵𝐵))⊆�𝐹𝐹_𝐵𝐵.

Proof: Let 𝐹𝐹_𝐵𝐵 be a soft pre-closed set. Then (𝐹𝐹_𝐵𝐵)𝑐𝑐 is a soft pre-open set. So (𝐹𝐹_𝐵𝐵)𝑐𝑐⊆� int(cl((𝐹𝐹_𝐵𝐵)𝑐𝑐))⇒ (𝐹𝐹_𝐵𝐵)𝑐𝑐⊆� [cl(int(𝐹𝐹_𝐵𝐵))]c which implies that (𝐹𝐹_𝐵𝐵)⊇�cl(int(𝐹𝐹_𝐶𝐶)). Conversely, cl(int(𝐹𝐹_𝐵𝐵))⊆� 𝐹𝐹_𝐵𝐵⇒[cl(int(𝐹𝐹_𝐵𝐵))] c⊇� (𝐹𝐹_𝐵𝐵)𝑐𝑐 which implies that int(cl(𝐹𝐹_𝐵𝐵) c)] ⊇� (𝐹𝐹_𝐵𝐵)𝑐𝑐. Therefore, 𝐹𝐹_𝐵𝐵𝑐𝑐 is a soft pre-open set and so 𝐹𝐹_𝐵𝐵 is a soft pre-closed set.

Theorem 3.11: Let (𝐹𝐹_𝐴𝐴 ,𝜏𝜏̃) be a soft topological space in which every soft subset is pre-open if and only if every soft open set in (𝐹𝐹_𝐴𝐴 ,τ�) is soft closed set.

Proof: Suppose that every soft open set is a soft closed set. Let 𝐹𝐹_𝐵𝐵 be a soft set. Let 𝐹𝐹_𝐶𝐶 =𝐹𝐹_𝐵𝐵𝑐𝑐 and so, 𝐹𝐹_𝐵𝐵 =𝐹𝐹_𝐶𝐶𝑐𝑐. Since int(𝐹𝐹_𝐵𝐵) is a soft open set, by hypothesis, int(𝐹𝐹_𝐵𝐵) is a soft closed set and so cl(int(𝐹𝐹_𝐵𝐵))=int(𝐹𝐹_𝐵𝐵)⊆� 𝐹𝐹_𝐵𝐵=𝐹𝐹_𝐶𝐶𝑐𝑐.

Then 𝐹𝐹_𝐶𝐶⊆�(cl(int(𝐹𝐹_𝐵𝐵))) c = int(cl(𝐹𝐹_𝐵𝐵𝑐𝑐)) = int(cl(𝐹𝐹_𝐶𝐶)). Every soft subset is soft pre-open set. Conversely, let 𝐹𝐹_𝐵𝐵 be a soft open set. Since 𝐹𝐹_𝐵𝐵𝑐𝑐 is a soft set, by hypothesis, 𝐹𝐹_𝐵𝐵𝑐𝑐⊆� int(cl(𝐹𝐹_𝐵𝐵𝑐𝑐)) and so cl(int(𝐹𝐹_𝐵𝐵))⊆�𝐹𝐹_𝐵𝐵 which implies that cl(𝐹𝐹_𝐵𝐵)⊆� 𝐹𝐹_𝐵𝐵.

Therefore, every soft open set is a soft closed set. This complete the proof.

Definition 3.12: Let (𝐹𝐹_𝐴𝐴 ,τ�) be a soft topological space and 𝐹𝐹_𝐵𝐵 be a soft subset of 𝐹𝐹_𝐴𝐴.

(i) The soft pre-interior of.𝐹𝐹_𝐵𝐵 is the soft set p-int(𝐹𝐹_𝐵𝐵) =Ũ {𝐹𝐹_𝐶𝐶: 𝐹𝐹_𝐶𝐶 is soft pre-open set and 𝐹𝐹_𝐶𝐶

⊆

�𝐹𝐹_𝐵𝐵}. (ii) The soft pre-closure of 𝐹𝐹_𝐵𝐵is the soft set p-cl(𝐹𝐹_𝐵𝐵) =∩�{𝐹𝐹_𝐶𝐶: 𝐹𝐹_𝐶𝐶is soft pre-closed set and 𝐹𝐹_𝐵𝐵

⊆

�𝐹𝐹_𝐶𝐶}.

Its clearly, p-int(𝐹𝐹_𝐵𝐵) is the largest soft pre-open set contained in 𝐹𝐹_𝐵𝐵 and p-cl(𝐹𝐹_𝐵𝐵) is the smallest soft pre-closed set containing 𝐹𝐹_𝐵𝐵.

Theorem 3.13: Let (𝐹𝐹_𝐴𝐴 ,τ�) be a soft topological space and 𝐹𝐹_𝐵𝐵 be a soft subset of 𝐹𝐹_𝐴𝐴. Then the following holds.

(i) (p-cl(𝐹𝐹_𝐵𝐵))𝑐𝑐 = p-int(𝐹𝐹_𝐵𝐵𝑐𝑐) (ii) (p-int(𝐹𝐹_𝐵𝐵))𝑐𝑐 = p-cl(𝐹𝐹_𝐵𝐵𝑐𝑐) (iii)(p-int(𝐹𝐹_𝐵𝐵))𝑐𝑐 = p-cl(𝐹𝐹_𝐵𝐵𝑐𝑐)

Proof:

(i) (p-cl(𝐹𝐹_𝐵𝐵))𝑐𝑐= (∩�{𝐹𝐹_𝐶𝐶: 𝐹𝐹_𝐶𝐶 is soft pre-closed set and 𝐹𝐹_𝐶𝐶⊆� 𝐹𝐹_𝐵𝐵})𝑐𝑐 = Ũ{𝐹𝐹_𝐶𝐶𝑐𝑐: 𝐹𝐹_𝐶𝐶𝑐𝑐 is soft pre-open set and 𝐹𝐹_𝐶𝐶𝑐𝑐⊆�𝐹𝐹_𝐵𝐵𝑐𝑐}

= Ũ{𝐹𝐹_𝐶𝐶𝑐𝑐: 𝐹𝐹_𝐶𝐶𝑐𝑐 is soft pre-open set and 𝐹𝐹_𝐶𝐶𝑐𝑐⊆�𝐹𝐹_𝐵𝐵𝑐𝑐}= p-int(𝐹𝐹_𝐵𝐵𝑐𝑐). (ii) (s-int(𝐹𝐹_𝐵𝐵))𝑐𝑐 = (Ũ{𝐹𝐹_𝐶𝐶: 𝐹𝐹_𝐶𝐶 is soft pre-open set and 𝐹𝐹_𝐶𝐶⊆�𝐹𝐹_𝐵𝐵})𝑐𝑐

= ∩�{𝐹𝐹_𝐶𝐶𝑐𝑐 : 𝐹𝐹_𝐶𝐶𝑐𝑐 is soft pre-open set and 𝐹𝐹_𝐵𝐵𝑐𝑐⊆�𝐹𝐹_𝐶𝐶𝑐𝑐 }

= ∩�{𝐹𝐹_𝐶𝐶𝑐𝑐: 𝐹𝐹_𝐶𝐶𝑐𝑐 is soft pre-closed set and 𝐹𝐹_𝐵𝐵𝑐𝑐⊆�𝐹𝐹_𝐶𝐶𝑐𝑐 }= p-cl(𝐹𝐹_𝐵𝐵𝑐𝑐).

(iii) The proof follows from (ii). □

Theorem 3.14: Let (F_A ,τ�) be a soft topological space and let 𝐹𝐹_𝐵𝐵 and 𝐹𝐹_𝐶𝐶 be soft subsets of 𝐹𝐹_𝐴𝐴. Then the following holds.

(i) p-cl(𝐹𝐹_𝜙𝜙 ) = 𝐹𝐹_𝜙𝜙 and p-cl(𝐹𝐹_𝐴𝐴 ) = 𝐹𝐹_𝐴𝐴.

(ii) 𝐹𝐹_𝐵𝐵 is a soft pre-closed set if and only if 𝐹𝐹_𝐵𝐵=p-cl(𝐹𝐹_𝐵𝐵). (iii)p-cl(p-cl(𝐹𝐹_𝐵𝐵)) = p-cl(𝐹𝐹_𝐵𝐵).

(iv)𝐹𝐹_𝐵𝐵⊆�𝐹𝐹_𝐶𝐶 implies p-cl(𝐹𝐹_𝐵𝐵)⊆�p-cl(𝐹𝐹_𝐶𝐶). (v) p-cl(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶)⊆�p-cl(𝐹𝐹_𝐵𝐵)∩�p-cl(𝐹𝐹_𝐶𝐶). (vi)p-cl(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶) = p-cl(𝐹𝐹_𝐵𝐵)Ũp-cl(𝐹𝐹_𝐶𝐶).

Proof:

(i) is obvious.

(ii) If 𝐹𝐹_𝐵𝐵 is soft pre-closed set, then 𝐹𝐹_𝐵𝐵 is itself a soft pre-closed set in 𝐹𝐹_𝐴𝐴 which contains 𝐹𝐹_𝐵𝐵 . Since p-cl(𝐹𝐹_𝐵𝐵) is the smallest soft pre-closed set containing 𝐹𝐹_𝐵𝐵, and 𝐹𝐹_𝐵𝐵 = p-cl(𝐹𝐹_𝐵𝐵). Conversely, suppose that 𝐹𝐹_𝐵𝐵 = p-cl(𝐹𝐹_𝐵𝐵). Since p-cl(𝐹𝐹_𝐵𝐵) being the intersection of soft pre-closed set is soft pre-closed set, 𝐹𝐹_𝐵𝐵 is soft pre-closed sets in 𝐹𝐹_𝐴𝐴. (iii)Since p-cl(𝐹𝐹_𝐵𝐵) is a soft pre-closed set, by part (ii). Therefore, p-cl(p-cl(𝐹𝐹_𝐵𝐵)) = p-cl(𝐹𝐹_𝐵𝐵).

(iv)Suppose that 𝐹𝐹_𝐵𝐵

⊆

� 𝐹𝐹_𝐶𝐶 . p-cl(𝐹𝐹_𝐵𝐵)= (∩�{𝐹𝐹_𝐷𝐷 : 𝐹𝐹_𝐵𝐵

⊆

�𝐹𝐹_𝐷𝐷 and 𝐹𝐹_𝐷𝐷 belongs to soft pre-closed set in 𝐹𝐹_𝐴𝐴} and p-cl(𝐹𝐹_𝐶𝐶)= (∩�{𝐹𝐹_𝐸𝐸: 𝐹𝐹_𝐶𝐶

⊆

�𝐹𝐹_𝐸𝐸 and 𝐹𝐹_𝐸𝐸 belongs to soft pre-closed set in 𝐹𝐹_𝐴𝐴}. Since 𝐹𝐹_𝐵𝐵

⊆

� p-cl(𝐹𝐹_𝐵𝐵) and 𝐹𝐹_𝐶𝐶

⊆

� p-cl(𝐹𝐹_𝐶𝐶)

⇒

𝐹𝐹_𝐵𝐵

⊆

�𝐹𝐹_𝐶𝐶

⊆

�

p-cl(𝐹𝐹_𝐶𝐶)

⇒

𝐹𝐹_𝐵𝐵

⊆

� p-cl(𝐹𝐹_𝐶𝐶). But p-cl(𝐹𝐹_𝐵𝐵) is the smallest soft pre-closed set containing𝐹𝐹_𝐵𝐵. Therefore,

(v) Since 𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶

⊆

� 𝐹𝐹_𝐵𝐵 and 𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶

⊆

� 𝐹𝐹_𝐶𝐶, by part (iv), p-cl(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶)

⊆

� p-cl(𝐹𝐹_𝐵𝐵) and

p-cl(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶)

⊆

� p-cl(𝐹𝐹_𝐶𝐶) . Thus p-cl(𝐹𝐹_𝐵𝐵 ∩� 𝐹𝐹_𝐶𝐶)

⊆

� p-cl(𝐹𝐹_𝐵𝐵) ∩� p-cl(𝐹𝐹_𝐶𝐶).

(vii)Since 𝐹𝐹_𝐵𝐵

⊆

� 𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶 and 𝐹𝐹_𝐶𝐶

⊆

�𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶, by part (iv). Then p-cl(𝐹𝐹_𝐵𝐵)

⊆

� p-cl(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶) and p-cl(𝐹𝐹_𝐶𝐶)

⊆

�p-cl(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶),

which implies that p-cl(𝐹𝐹_𝐵𝐵)Ũ p-cl(𝐹𝐹_𝐶𝐶)

⊆

� p-cl(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶). Now, p-cl(𝐹𝐹_𝐵𝐵) and p-cl(𝐹𝐹_𝐶𝐶) are soft pre-closed sets in 𝐹𝐹𝐴𝐴, which implies that p-cl(𝐹𝐹𝐵𝐵)Ũ p-cl(𝐹𝐹𝐶𝐶) is a soft pre-closed set in 𝐹𝐹𝐴𝐴. Then 𝐹𝐹𝐵𝐵

⊆

�p-cl(𝐹𝐹𝐵𝐵) and 𝐹𝐹𝐶𝐶

⊆

� p-cl(𝐹𝐹𝐶𝐶)

imply that 𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶

⊆

� p-cl(𝐹𝐹_𝐵𝐵)Ũp-cl(𝐹𝐹_𝐶𝐶). That is p-cl(𝐹𝐹_𝐵𝐵)Ũ p-cl(𝐹𝐹_𝐶𝐶) is a soft pre-closed set containing 𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶. But p-cl(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶) is the smallest soft pre-closed set containing 𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶.

Hence p-cl(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶)⊆� p-cl(𝐹𝐹_𝐵𝐵)Ũ p-cl(𝐹𝐹_𝐶𝐶). Therefore, p-cl(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶) = p-cl(𝐹𝐹_𝐵𝐵 )Ũ p-cl(𝐹𝐹_𝐶𝐶). □

Theorem 3.15: Let (𝐹𝐹_𝐴𝐴 ,τ�) be a soft topological space and let 𝐹𝐹_𝐵𝐵 and 𝐹𝐹_𝐶𝐶 be soft subsets of 𝐹𝐹_𝐴𝐴. Then the following holds.

(i) p-int(𝐹𝐹_𝜙𝜙 ) = 𝐹𝐹_𝜙𝜙 and p-int(𝐹𝐹_𝐴𝐴) = 𝐹𝐹_𝐴𝐴.

(ii) 𝐹𝐹_𝐵𝐵 is a soft pre-open set if and only if 𝐹𝐹_𝐵𝐵 = p-int(𝐹𝐹_𝐵𝐵).

(iii)p-int(p-int(𝐹𝐹_𝐵𝐵))=p-int(𝐹𝐹_𝐵𝐵).

(iv)𝐹𝐹_𝐵𝐵⊆�𝐹𝐹_𝐶𝐶 implies p-int(𝐹𝐹_𝐵𝐵)⊆� p-int(𝐹𝐹_𝐶𝐶).

(v) p-int(𝐹𝐹_𝐵𝐵)∩�p-int(𝐹𝐹_𝐶𝐶)⊆� p-int(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶).

(vi)p-int(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶) = p-int(𝐹𝐹_𝐵𝐵)Ũ p-int(𝐹𝐹_𝐶𝐶).

Proof:

(i) is obvious.

(ii) If 𝐹𝐹_𝐵𝐵 is soft pre-open set, then 𝐹𝐹_𝐵𝐵 is itself a soft pre-open set in 𝐹𝐹_𝐴𝐴 which contains 𝐹𝐹_𝐵𝐵. So p-int(𝐹𝐹_𝐵𝐵) is the largest soft pre-open set contained in 𝐹𝐹_𝐵𝐵 and 𝐹𝐹_𝐵𝐵= s-int(𝐹𝐹_𝐵𝐵). Conversely, suppose that 𝐹𝐹_𝐵𝐵= p-int(𝐹𝐹_𝐵𝐵). Since

p-int(𝐹𝐹_𝐵𝐵) being the union of soft pre-open sets is soft pre-open sets, so p-int(𝐹𝐹_𝐵𝐵) be a soft pre-open set of 𝐹𝐹_𝐴𝐴 which implies that 𝐹𝐹_𝐵𝐵 is soft pre-open set in 𝐹𝐹_𝐴𝐴.

(iii)Since p-int(𝐹𝐹_𝐵𝐵) is a soft pre-open set, by part (ii). Therefore, p-int(p-int(𝐹𝐹_𝐵𝐵)) = p-int(𝐹𝐹_𝐵𝐵).

(iv)Suppose that 𝐹𝐹_𝐵𝐵

⊆

� 𝐹𝐹_𝐶𝐶. p-int(𝐹𝐹_𝐵𝐵)= Ũ{𝐹𝐹_𝐷𝐷 : 𝐹𝐹_𝐷𝐷

⊆

�𝐹𝐹_𝐵𝐵 and 𝐹𝐹_𝐷𝐷 be a soft pre-open set in 𝐹𝐹_𝐴𝐴} and p-int(𝐹𝐹_𝐶𝐶)= Ũ {𝐹𝐹_𝐸𝐸:

𝐹𝐹𝐸𝐸

⊆

�𝐹𝐹𝐶𝐶 and 𝐹𝐹𝐸𝐸 be a soft pre-open set in 𝐹𝐹𝐴𝐴}. Since p-int(𝐹𝐹𝐵𝐵)

⊆

� 𝐹𝐹𝐵𝐵

⊆

� 𝐹𝐹𝐶𝐶, p-int(𝐹𝐹𝐵𝐵)

⊆

�𝐹𝐹𝐶𝐶. Since p-int(𝐹𝐹𝐶𝐶) is

the largest soft pre-open set contained in 𝐹𝐹_𝐶𝐶, p-int(𝐹𝐹_𝐵𝐵)

⊆

� p-int(𝐹𝐹_𝐶𝐶).

(v) Since 𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶

⊆

�𝐹𝐹_𝐵𝐵 and 𝐹𝐹_𝐵𝐵 ∩� 𝐹𝐹_𝐶𝐶

⊆

�𝐹𝐹_𝐶𝐶, by part (iv), 𝐹𝐹_𝐵𝐵

⊆

�𝐹𝐹_𝐶𝐶 which implies that p-int(𝐹𝐹_𝐵𝐵)

⊆

� p-int(𝐹𝐹_𝐶𝐶). Since

p-int(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶)

⊆

� p-int(𝐹𝐹_𝐵𝐵) and p-int(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶)

⊆

� p-int(𝐹𝐹_𝐶𝐶) which implies that p-int(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶)

⊆

� p-int(𝐹𝐹_𝐵𝐵)∩� p-int(𝐹𝐹_𝐶𝐶). Now, p-int(𝐹𝐹_𝐵𝐵) and p-int(𝐹𝐹_𝐶𝐶) are soft pre-open sets in 𝐹𝐹_𝐴𝐴. So that p-int(𝐹𝐹_𝐵𝐵)∩�p-int(𝐹𝐹_𝐶𝐶) be a soft pre-open set in 𝐹𝐹_𝐴𝐴.Then p-int(𝐹𝐹_𝐵𝐵)

⊆

� 𝐹𝐹_𝐵𝐵and p-int(𝐹𝐹_𝐶𝐶)

⊆

�𝐹𝐹_𝐶𝐶 which implies that p-int(𝐹𝐹_𝐵𝐵)∩�p-int(𝐹𝐹_𝐶𝐶)

⊆

� p-int(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶). That is p-int(𝐹𝐹_𝐵𝐵)∩�p-int(𝐹𝐹_𝐶𝐶) is a soft pre-open set contained in 𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶. But, p-int(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶) is the largest soft pre-open set contained in 𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶. Hence p-int(𝐹𝐹_𝐵𝐵)∩�p-int(𝐹𝐹_𝐶𝐶)

⊆

� p-int(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶). Therefore, p-int(𝐹𝐹_𝐵𝐵∩� 𝐹𝐹_𝐶𝐶)= p-int(𝐹𝐹_𝐵𝐵)∩�p-int(𝐹𝐹_𝐶𝐶).

(vi)Since 𝐹𝐹_𝐵𝐵

⊆

� 𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶 and 𝐹𝐹_𝐶𝐶

⊆

� 𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶, by part (iv), 𝐹𝐹_𝐵𝐵

⊆

�𝐹𝐹_𝐶𝐶 which implies that p-int(𝐹𝐹_𝐵𝐵)

⊆

� p-int(𝐹𝐹_𝐶𝐶). Then

p-int(𝐹𝐹_𝐵𝐵)

⊆

� p-int(𝐹𝐹_𝐵𝐵 Ũ𝐹𝐹_𝐶𝐶) and p-int(𝐹𝐹_𝐶𝐶)

⊆

� p-int(𝐹𝐹_𝐵𝐵 Ũ𝐹𝐹_𝐶𝐶), which implies that p-int(𝐹𝐹_𝐵𝐵)Ũp-int(𝐹𝐹_𝐶𝐶)⊆� p-int(𝐹𝐹_𝐵𝐵

Ũ𝐹𝐹𝐶𝐶). Now, p-int(𝐹𝐹𝐵𝐵) and p-int(𝐹𝐹𝐶𝐶) are soft pre-open sets in 𝐹𝐹𝐴𝐴 which implies that p-int(𝐹𝐹𝐵𝐵)Ũ p-int(𝐹𝐹𝐶𝐶) is a

soft pre-open set in 𝐹𝐹_𝐴𝐴. Then 𝐹𝐹_𝐵𝐵⊆� p-int(𝐹𝐹_𝐵𝐵) and 𝐹𝐹_𝐶𝐶⊆� p-int(𝐹𝐹_𝐶𝐶) imply that 𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶⊆� p-int(𝐹𝐹_𝐵𝐵)Ũp-int(𝐹𝐹_𝐶𝐶). That is p-int(𝐹𝐹_𝐵𝐵)Ũ s-int(𝐹𝐹_𝐶𝐶) is a soft pre-open set containing 𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶. Hence p-int(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶)⊆� p-int(𝐹𝐹_𝐵𝐵)Ũ p-int(𝐹𝐹_𝐶𝐶). Therefore, p-int(𝐹𝐹_𝐵𝐵Ũ𝐹𝐹_𝐶𝐶)=p-int(𝐹𝐹_𝐵𝐵)Ũ p- int(𝐹𝐹_𝐶𝐶). □

Theorem 3.16: Let (𝐹𝐹_𝐴𝐴 ,τ�) be a soft topological space and let 𝐹𝐹_𝐵𝐵 be soft subset of F_A. Then the following are equivalent.

(i) 𝐹𝐹_𝐵𝐵 is a soft pre-closed set

(ii) int(cl(𝐹𝐹_𝐵𝐵))⊆�𝐹𝐹_𝐵𝐵 (iii)cl(int(𝐹𝐹_𝐵𝐵𝑐𝑐))⊇� 𝐹𝐹_𝐵𝐵𝑐𝑐

(iv)𝐹𝐹_𝐵𝐵𝑐𝑐 is a soft pre-open set.

Proof:

(i) ⇒ (ii): If 𝐹𝐹_𝐵𝐵 is a soft pre-closed set, then there exist soft closed set 𝐹𝐹_𝐶𝐶 such that int(𝐹𝐹_𝐶𝐶)⊆�𝐹𝐹_𝐵𝐵⊆� 𝐹𝐹_𝐶𝐶⇒int(𝐹𝐹_𝐶𝐶)⊆�𝐹𝐹_𝐵𝐵

⊆_�cl(𝐹𝐹_𝐵𝐵 )⊆�𝐹𝐹_𝐶𝐶. By the property of interior, we then have int(cl(𝐹𝐹_𝐵𝐵))⊆� int(𝐹𝐹_𝐶𝐶)⊆�𝐹𝐹_𝐵𝐵.

(ii)⇒(iii): int(cl(𝐹𝐹_𝐵𝐵))⊆� 𝐹𝐹_𝐵𝐵⇒𝐹𝐹_𝐵𝐵𝑐𝑐⊆� int(cl(𝐹𝐹_𝐵𝐵))𝑐𝑐 =cl(int(𝐹𝐹_𝐵𝐵𝑐𝑐)).

(iii)⇒(iv): int(𝐹𝐹_𝐵𝐵𝑐𝑐) is an soft open set such that int(𝐹𝐹_𝐵𝐵𝑐𝑐)⊆�𝐹𝐹_𝐵𝐵𝑐𝑐⊆� cl(int(𝐹𝐹_𝐵𝐵𝑐𝑐)). Hence 𝐹𝐹_𝐵𝐵𝑐𝑐 is a soft pre-open set.

(iv)⇒(i): Given that 𝐹𝐹_𝐵𝐵𝑐𝑐 is a soft pre-open set. Therefore, (𝐹𝐹_𝐵𝐵𝑐𝑐)𝑐𝑐= 𝐹𝐹_𝐵𝐵 is a soft pre-closed set. □

Theorem 3.17: Let 𝐹𝐹_𝐵𝐵 be a soft subset of a soft topological space (𝐹𝐹_𝐴𝐴 ,τ�). Then the following holds. (i) p-cl(𝐹𝐹_𝐵𝐵)=𝐹𝐹_𝐵𝐵Ũ cl(int(𝐹𝐹_𝐵𝐵))

Proof:

(i) cl[int[𝐹𝐹_𝐵𝐵Ũ cl(int(𝐹𝐹_𝐵𝐵))]]⊆� cl[int(𝐹𝐹_𝐵𝐵)Ũ cl(int(𝐹𝐹_𝐵𝐵))]=cl(int(𝐹𝐹_𝐵𝐵))⊆�𝐹𝐹_𝐵𝐵Ũ cl(int(𝐹𝐹_𝐵𝐵)) by Lemma 2.18. Hence 𝐹𝐹_𝐵𝐵Ũ

cl(int(𝐹𝐹_𝐵𝐵)) is a soft pre-closed set and thus p-cl(𝐹𝐹_𝐵𝐵)⊆�𝐹𝐹_𝐵𝐵Ũ cl(int(𝐹𝐹_𝐵𝐵)). On the other hand, since p-cl(𝐹𝐹_𝐵𝐵) is a soft pre-closed set, we have cl(int(𝐹𝐹_𝐵𝐵))⊆� cl(int(p-cl(𝐹𝐹_𝐵𝐵)))⊆� p-cl(𝐹𝐹_𝐵𝐵). Hence 𝐹𝐹_𝐵𝐵Ũ cl(int(𝐹𝐹_𝐵𝐵))⊆� p-cl(𝐹𝐹_𝐵𝐵) which implies that p-cl(𝐹𝐹_𝐵𝐵)=𝐹𝐹_𝐵𝐵Ũ cl(int(𝐹𝐹_𝐵𝐵)).

(ii) is a consequence of (i). □

Theorem 3.18: Let 𝐹𝐹_𝐵𝐵 be a soft subset of a soft topological space (𝐹𝐹_𝐴𝐴 ,τ�). Then the following holds.

(i) cl(p-int(𝐹𝐹_𝐵𝐵)) = cl(int(cl(𝐹𝐹_𝐵𝐵)))

(ii) p-cl(p-int(𝐹𝐹_𝐵𝐵)) = p-int(𝐹𝐹_𝐵𝐵)Ũ cl(int(𝐹𝐹_𝐵𝐵)).

Proof:

(i) Let cl(p-int(𝐹𝐹_𝐵𝐵))=cl[𝐹𝐹_𝐵𝐵∩� int(cl(𝐹𝐹_𝐵𝐵))] by Lemma 2.18.⊆�cl(𝐹𝐹_𝐵𝐵)∩� cl(int(cl(𝐹𝐹_𝐵𝐵)))⊆�cl(int(cl(𝐹𝐹_𝐵𝐵))). Now,

cl(int(cl(𝐹𝐹_𝐵𝐵)))⊆� cl[int(cl(𝐹𝐹_𝐵𝐵)))∩� cl(𝐹𝐹_𝐵𝐵)]⊆� cl(cl(int(cl(𝐹𝐹_𝐵𝐵)))∩� 𝐹𝐹_𝐵𝐵) ⊆� cl(int(cl(𝐹𝐹_𝐵𝐵))∩� 𝐹𝐹_𝐵𝐵) [by Lemma 2.18.] ⊆� cl(p-int(𝐹𝐹_𝐵𝐵)). Hence cl(int(cl(𝐹𝐹_𝐵𝐵)))= cl(p-int(𝐹𝐹_𝐵𝐵)).

(ii) Since soft open set is a soft pre-open set, we have intl(𝐹𝐹_𝐵𝐵)⊆�p-int(𝐹𝐹_𝐵𝐵) ⊆�𝐹𝐹_𝐵𝐵. Therefore, int(p-int(𝐹𝐹_𝐵𝐵))=int(𝐹𝐹_𝐵𝐵).

Now, by Theorem 3.17(ii) p-cl(p-int(𝐹𝐹_𝐵𝐵))= p-int(𝐹𝐹_𝐵𝐵)Ũ cl(int(p-int(𝐹𝐹_𝐵𝐵))) p-int(𝐹𝐹_𝐵𝐵)Ũ cl(int(𝐹𝐹_𝐵𝐵)).

Theorem 3.19: Let (𝐹𝐹_𝐴𝐴 ,τ�) be a soft topological space. Then for any 𝐹𝐹_𝐵𝐵⊆�𝐹𝐹_𝐴𝐴, p-int(𝐹𝐹_𝐵𝐵)=𝐹𝐹_𝐴𝐴−p-cl(𝐹𝐹_𝐴𝐴− 𝐹𝐹_𝐵𝐵).

Proof: Let 𝐹𝐹_𝐴𝐴−p-int(𝐹𝐹_𝐵𝐵)=𝐹𝐹_𝐴𝐴−[𝐹𝐹_𝐵𝐵∩� int(cl(𝐹𝐹_𝐵𝐵))]=[𝐹𝐹_𝐴𝐴− 𝐹𝐹_𝐵𝐵]Ũ[𝐹𝐹_𝐴𝐴−int(cl(𝐹𝐹_𝐵𝐵))] = [𝐹𝐹_𝐴𝐴− 𝐹𝐹_𝐵𝐵]Ũ cl[𝐹𝐹_𝐴𝐴−cl(𝐹𝐹_𝐵𝐵)]= [𝐹𝐹_𝐴𝐴− 𝐹𝐹_𝐵𝐵]Ũ cl(int(𝐹𝐹_𝐴𝐴− 𝐹𝐹_𝐵𝐵)]=p-cl[𝐹𝐹_𝐴𝐴− 𝐹𝐹_𝐵𝐵], by Proposition 2.19 and Theorem 3.17. Therefore, p-int(𝐹𝐹_𝐵𝐵)=𝐹𝐹_𝐴𝐴− p-cl(𝐹𝐹_𝐴𝐴− 𝐹𝐹_𝐵𝐵). □

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Source of support: Nil, Conflict of interest: None Declared.