# Infra Soft Semiopen Sets and Infra Soft Semicontinuity

Authors: Tareq M. Al-shami; A. A. Azzam
Journal: Journal of Function Spaces
 (2021)

---

## Abstract

To contribute to the area of infra soft topology, we introduce one of the generalizations of infra soft open sets called infra soft semiopen sets. We establish some characterizations of them and study their main properties. We determine under what condition this class is closed under finite intersection and show that this class is preserved under infra soft continuous mappings and finite product of soft spaces. Then, we present the concepts of infra semi-interior, infra semiclosure, infra semilimit, and infra semiboundary soft points of a soft set and elucidate the relationships between them. Finally, we exploit infra soft semiopen and infra soft semiclosed sets to define new types of soft mappings. We characterize each one of these soft mappings and explore main features.


---

## Body

## 1. Introduction

In 1999, Molodtsov [1] presented a novel mathematical tool to address vagueness, namely, soft sets. He discussed its relationship with fuzzy sets and showed some applications in different fields. Then, many scholars and researchers have studied some applications of soft set in different scopes such as decision-making problems [2], computer science [3], and medical science [4].In 2003, Maji et al. [5] began studying the main concepts and notions of soft set theory. They explored the intersection and union operators, difference of two soft sets, and a complement of a soft set. However, some shortcomings appeared in their definitions, which led to reformulate most of these definitions and present new kinds of them. Ali et al. [6] originated new operators and operations between to preserve some properties and results of the (crisp) set theory in the soft set theory. Attempts were still in this path to produce new operators and relations like those introduced in [7].In 2011, Çag˘man et al. [8] and Shabir and Naz [9] made use of soft sets to define soft topological spaces. Whereas, Çag˘man et al.’s definition given over an absolute soft set and different sets of parameters, Shabir and Naz’ definition given over a fixed set of universe and a fixed set of parameters. This paper follows the definition of Shabir and Naz. Later on, many studies which investigated the topological concepts in soft topologies have been done such as soft compactness [10], soft connectedness [11], soft separation axioms, soft basis [12], Caliber and chain conditions [13], soft bioperators, and generalized soft open sets [14]. Also, uniformity and Menger structures were introduced in the context of soft sets in [15, 16], respectively.Soft topology was generalized to some structures; one of them is an infra soft topology [17]. The motivations of continuously investigating infra soft topological structure are that many topological properties are kept in the frame of infra soft topologies as well as the easy construction of examples that illustrate the relationships among the topological concepts. This matter was investigated for the concepts of infra soft compactness and infra soft connectedness in [18, 19].Generalizations of (soft) open sets are a major topic in (soft) topology. One of the important generalizations is a soft semiopen set [20] which was studied in classical topology by Levine [21]. In this article, we aim to explore the properties of this type of generalizations in the frame of infra soft topology. We elucidate the soundness of several properties of semiopen sets via infra soft topological spaces. This means that the infra soft topological spaces are flexible area to discuss the topological ideas and explore the relationships between them.The arrangement of this article is as follows: Section2 is allocated to mention some definitions and results relating to soft set theory and infra soft topology. In Section 3, we define a class of infra soft semiopen sets and establish some of characterizes. The concepts of infra semi-interior, infra semiclosure, infra semilimit, and infra semiboundary soft points of a soft set are introduced and probed in Section 4. In Section 5, we study the concepts of infra soft semicontinuous, infra soft semiopen, infra soft semiclosed, and infra soft semihomeomorphism mappings. Also, we formulate and study the concept of semifixed soft points in the frame of infra soft topologies. Finally, some conclusions and the possible upcoming works are given in Section 6.


## 2. Preliminaries

This section mentions the concepts and findings that we need to understand this manuscript.


### 2.1. Soft Set Theory

Definition 1 (see [1]).
ConsiderΘ as a set of parameters, T a universal set, and 2T the power set of T. An ordered pair Ω,Θ is called a soft set over T provided that Ω:Θ⟶2T is a crisp mapping. A soft set is expressed as Ω,Θ=θ,Ωθ:θ∈ΘandΩθ∈2T. We call Ωθ a θ-approximate of Ω,Θ.
The class of all soft sets overT under a set of parameters Θ is symbolized by CTΘ.Definition 2 (see [6]).
The complement of a soft setΩ,Θ, denoted by Ωc,Θ, provided that a mapping Ωc:Θ⟶2T is given by Ωcθ=T\Ωθ for each θ∈Θ.Definition 3 (see [5]).
We callΩ,Θ an absolute (resp., a null) soft set over T if the image of each parameter of Θ under a mapping Ω:Θ⟶2T is the universal set T (resp., empty set).
The absolute and null soft sets are symbolized byT~ and Φ, respectively.Definition 4 (see [22]).
We callΩ,Θ a stable (resp., finite, countable) soft set if every θ-approximate of Ω,Θ is equal (resp., finite, countable). Otherwise, it is called unstable (resp., infinite, uncountable).Definition 5 (see [23]).
ConsiderΩ,Θ as a soft set over T such that Ωθ=t∈T and Ωθ′=∅ for each θ′≠θ. Then, we call Ω,Θ a soft point over T. It will be symbolized by δθt.Definition 6 (see [6]).
The intersection of soft setsΩ,Θ and Ψ,Δ which are defined over T, symbolized by Ω,Θ∩~Ψ,Δ, is a soft set Y,Σ, where Σ=Θ∩Δ≠∅, and a mapping Y:Σ⟶2T is given by Yθ=Ωθ∩Ψθ for each θ∈Σ.Definition 7 (see [5]).
The union of soft setsΩ,Θ and Ψ,Δ which are defined over T, symbolized by Ω,Θ∪~Ψ,Δ, is a soft set Y,Σ, where Σ=Θ∪Δ and a mapping Y:Σ⟶2T is given as follows:
(1)Yθ=Ωθθ∈Θ\Δ,Ψθθ∈Δ\Θ,Ωθ∪Ψθθ∈Θ∩Δ.Definition 8 (see [24]).
A soft setΩ,Θ is a subset of a soft set Ψ,Δ, symbolized by Ω,Θ⊆~Ψ,Δ, if Θ⊆Δ and Ωθ⊆Ψθ for all θ∈Θ. The soft sets Ω,Θ and Ψ,Δ are called soft equal if Ω,Θ⊆~Ψ,Δ and Ψ,Δ⊆~Ω,Θ.Definition 9 (see [10]).
The Cartesian product ofΩ,Θ and Ψ,Δ, symbolized by Ω×Ψ,Θ×Δ, is defined as Ω×Ψθ,θ′=Ωθ×Ψθ′ for each θ,θ′∈Θ×Δ.The definition of soft mappings given in [25] was reformulated in a way that reduces calculation burden and gives a justification (logical explanation) for some soft concepts such as why we determine that Eτ is injective, or surjective according to its two crisp maps E and τ.Definition 10 (see [26]).
LetE:T⟶S and τ:Θ⟶Σ be two crisp mappings. A soft mapping Eτ of CTΘ into CSΣ is a relation such that each soft point in CTΘ is related to one and only one soft point in CSΣ such that
(2)Eτδθt=δτθEtforeachδθt∈CTΘ.
In addition,Eτ−1δσs=∪~ω∈τ−1σν∈E−1sδων for each δσs∈CSΣ.Definition 11 (see [25]).
A soft mappingfτ:CTΘ⟶CSΔ is said to be injective (resp., surjective, bijective) if both f and τ are injective (resp., surjective, bijective).


### 2.2. Infra Soft Topological Spaces

Definition 12 (see [17]).
A familyξ of soft sets over T with Θ as a parameter set is said to be an infra soft topology on T if it is closed under finite intersection and Φ is a member of ξ.The tripleT,ξ,Θ is called an infra soft topological space (briefly, ISTS). We call a member of ξ an infra soft open set and call its complement an infra soft closed set. We call T,ξ,Θ stable if all its infra soft open sets are stable.Definition 13 (see [17]).
LetΩ,Θ be a subset of T,ξ,Θ.
(i)
The intersection of all infra soft closed subsets ofT,ξ,Θ which contains a soft set Ω,Θ is called the infra soft closure of Ω,Θ. It is denoted by ClΩ,Θ(ii)
The union of all infra soft open subsets ofT,ξ,Θ which are contained in a soft set Ω,Θ is called the infra soft interior of Ω,Θ. It is denoted by IntΩ,ΘIt was shown in [17] that ClΩ,Θ and IntΩ,Θ need not be infra soft closed and infra soft open, respectively. Through this paper, Ω,Θ is called ξ-infra soft open (resp., ξ-infra soft closed) if IntΩ,Θ=Ω,Θresp.,ClΩ,Θ=Ω,Θ.Proposition 14 (see [17]).
LetΩ,Θ and Ψ,Θ subsets of an ISTS T,ξ,Θ. Then
(i)
ClΩ,Θ∪~Ψ,Θ=ClΩ,Θ∪~ClΨ,Θ(ii)
IntΩ,Θ∩~Ψ,Θ=IntΩ,Θ∩~IntΨ,ΘProposition 15 (see [17]).
LetΩ,Θ be an infra soft open set. Then
(3)Ω,Θ∩~ClΨ,Θ⊆~ClΩ,Θ∪~Ψ,ΘforanysubsetΨ,ΘofT,ξ,Θ.Proposition 16 (see [17]).
LetΩ,Θ be an infra soft closed set. Then
(4)IntΩ,Θ∪~Ψ,Θ⊆~Ω,Θ∩~IntΨ,ΘforanysubsetΩ,ΘofT,ξ,Θ.Definition 17.
A soft mappingfτ:T,ξ,Θ⟶S,π,Δ is said to be an infra soft homeomorphism if it is bijective, infra soft continuous (i.e., the image of every infra soft open set is infra soft open), and infra soft open (i.e., the image of every infra soft open set is infra soft open).A property is called an infra soft topological property (briefly, IST property) if it is preserved by any infra soft homeomorphism.Definition 18 (see [17]).
LetEτ:T,ξ,Θ⟶S,π,Δ be a soft mapping and M≠∅ be a subset of T. A soft mapping Eτ∣M:M,ξM,Θ⟶S,π,Δ which given by Eτ∣Mδθm=Eτδθm for every δθm∈M~ is called a restriction soft mapping of Eτ on M.Proposition 19.
LetTk,ξk,Θk:k∈K be a family of ISTSs. Then, ξ=∏k∈Kθk,Θk:θk,Θk∈τk is an infra soft topology on T=∏k∈KTk under a set of parameters B=∏k∈KΘk.We callξ given in proposition above, a product of infra soft topologies, and T,ξ,B a product of infra soft spaces.


## 2.1. Soft Set Theory

Definition 1 (see [1]).
ConsiderΘ as a set of parameters, T a universal set, and 2T the power set of T. An ordered pair Ω,Θ is called a soft set over T provided that Ω:Θ⟶2T is a crisp mapping. A soft set is expressed as Ω,Θ=θ,Ωθ:θ∈ΘandΩθ∈2T. We call Ωθ a θ-approximate of Ω,Θ.
The class of all soft sets overT under a set of parameters Θ is symbolized by CTΘ.Definition 2 (see [6]).
The complement of a soft setΩ,Θ, denoted by Ωc,Θ, provided that a mapping Ωc:Θ⟶2T is given by Ωcθ=T\Ωθ for each θ∈Θ.Definition 3 (see [5]).
We callΩ,Θ an absolute (resp., a null) soft set over T if the image of each parameter of Θ under a mapping Ω:Θ⟶2T is the universal set T (resp., empty set).
The absolute and null soft sets are symbolized byT~ and Φ, respectively.Definition 4 (see [22]).
We callΩ,Θ a stable (resp., finite, countable) soft set if every θ-approximate of Ω,Θ is equal (resp., finite, countable). Otherwise, it is called unstable (resp., infinite, uncountable).Definition 5 (see [23]).
ConsiderΩ,Θ as a soft set over T such that Ωθ=t∈T and Ωθ′=∅ for each θ′≠θ. Then, we call Ω,Θ a soft point over T. It will be symbolized by δθt.Definition 6 (see [6]).
The intersection of soft setsΩ,Θ and Ψ,Δ which are defined over T, symbolized by Ω,Θ∩~Ψ,Δ, is a soft set Y,Σ, where Σ=Θ∩Δ≠∅, and a mapping Y:Σ⟶2T is given by Yθ=Ωθ∩Ψθ for each θ∈Σ.Definition 7 (see [5]).
The union of soft setsΩ,Θ and Ψ,Δ which are defined over T, symbolized by Ω,Θ∪~Ψ,Δ, is a soft set Y,Σ, where Σ=Θ∪Δ and a mapping Y:Σ⟶2T is given as follows:
(1)Yθ=Ωθθ∈Θ\Δ,Ψθθ∈Δ\Θ,Ωθ∪Ψθθ∈Θ∩Δ.Definition 8 (see [24]).
A soft setΩ,Θ is a subset of a soft set Ψ,Δ, symbolized by Ω,Θ⊆~Ψ,Δ, if Θ⊆Δ and Ωθ⊆Ψθ for all θ∈Θ. The soft sets Ω,Θ and Ψ,Δ are called soft equal if Ω,Θ⊆~Ψ,Δ and Ψ,Δ⊆~Ω,Θ.Definition 9 (see [10]).
The Cartesian product ofΩ,Θ and Ψ,Δ, symbolized by Ω×Ψ,Θ×Δ, is defined as Ω×Ψθ,θ′=Ωθ×Ψθ′ for each θ,θ′∈Θ×Δ.The definition of soft mappings given in [25] was reformulated in a way that reduces calculation burden and gives a justification (logical explanation) for some soft concepts such as why we determine that Eτ is injective, or surjective according to its two crisp maps E and τ.Definition 10 (see [26]).
LetE:T⟶S and τ:Θ⟶Σ be two crisp mappings. A soft mapping Eτ of CTΘ into CSΣ is a relation such that each soft point in CTΘ is related to one and only one soft point in CSΣ such that
(2)Eτδθt=δτθEtforeachδθt∈CTΘ.
In addition,Eτ−1δσs=∪~ω∈τ−1σν∈E−1sδων for each δσs∈CSΣ.Definition 11 (see [25]).
A soft mappingfτ:CTΘ⟶CSΔ is said to be injective (resp., surjective, bijective) if both f and τ are injective (resp., surjective, bijective).


## 2.2. Infra Soft Topological Spaces

Definition 12 (see [17]).
A familyξ of soft sets over T with Θ as a parameter set is said to be an infra soft topology on T if it is closed under finite intersection and Φ is a member of ξ.The tripleT,ξ,Θ is called an infra soft topological space (briefly, ISTS). We call a member of ξ an infra soft open set and call its complement an infra soft closed set. We call T,ξ,Θ stable if all its infra soft open sets are stable.Definition 13 (see [17]).
LetΩ,Θ be a subset of T,ξ,Θ.
(i)
The intersection of all infra soft closed subsets ofT,ξ,Θ which contains a soft set Ω,Θ is called the infra soft closure of Ω,Θ. It is denoted by ClΩ,Θ(ii)
The union of all infra soft open subsets ofT,ξ,Θ which are contained in a soft set Ω,Θ is called the infra soft interior of Ω,Θ. It is denoted by IntΩ,ΘIt was shown in [17] that ClΩ,Θ and IntΩ,Θ need not be infra soft closed and infra soft open, respectively. Through this paper, Ω,Θ is called ξ-infra soft open (resp., ξ-infra soft closed) if IntΩ,Θ=Ω,Θresp.,ClΩ,Θ=Ω,Θ.Proposition 14 (see [17]).
LetΩ,Θ and Ψ,Θ subsets of an ISTS T,ξ,Θ. Then
(i)
ClΩ,Θ∪~Ψ,Θ=ClΩ,Θ∪~ClΨ,Θ(ii)
IntΩ,Θ∩~Ψ,Θ=IntΩ,Θ∩~IntΨ,ΘProposition 15 (see [17]).
LetΩ,Θ be an infra soft open set. Then
(3)Ω,Θ∩~ClΨ,Θ⊆~ClΩ,Θ∪~Ψ,ΘforanysubsetΨ,ΘofT,ξ,Θ.Proposition 16 (see [17]).
LetΩ,Θ be an infra soft closed set. Then
(4)IntΩ,Θ∪~Ψ,Θ⊆~Ω,Θ∩~IntΨ,ΘforanysubsetΩ,ΘofT,ξ,Θ.Definition 17.
A soft mappingfτ:T,ξ,Θ⟶S,π,Δ is said to be an infra soft homeomorphism if it is bijective, infra soft continuous (i.e., the image of every infra soft open set is infra soft open), and infra soft open (i.e., the image of every infra soft open set is infra soft open).A property is called an infra soft topological property (briefly, IST property) if it is preserved by any infra soft homeomorphism.Definition 18 (see [17]).
LetEτ:T,ξ,Θ⟶S,π,Δ be a soft mapping and M≠∅ be a subset of T. A soft mapping Eτ∣M:M,ξM,Θ⟶S,π,Δ which given by Eτ∣Mδθm=Eτδθm for every δθm∈M~ is called a restriction soft mapping of Eτ on M.Proposition 19.
LetTk,ξk,Θk:k∈K be a family of ISTSs. Then, ξ=∏k∈Kθk,Θk:θk,Θk∈τk is an infra soft topology on T=∏k∈KTk under a set of parameters B=∏k∈KΘk.We callξ given in proposition above, a product of infra soft topologies, and T,ξ,B a product of infra soft spaces.


## 3. Infra Soft Semiopen Sets and Basic Properties

In this section, we introduce the concept of infra soft semiopen sets which represents a class of generalizations of infra soft open sets. We give some characterizations of infra soft semiopen and infra soft semiclosed sets and establish main properties. Also, we prove that this class is closed under arbitrary unions and determine under what condition this class is closed under finite intersection. Finally, we show that an infra soft semiopen set and its complement are preserved under infra soft continuous mappings and finite product of soft spaces.Definition 20.
A subsetΩ,Θ of an ISTS T,ξ,Θ is said to be infra soft semiopen if Ω,Θ⊆~ClIntΩ,Θ. Its complement is said to be an infra soft semiclosed set.The following two propositions give some descriptions for infra soft semiopen and infra soft semiclosed sets.Proposition 21.
LetΩ,Θ be a subset of an ISTS T,ξ,Θ. Then, we have the following equivalent properties:
(i)
Ω,Θ is an infra soft semiopen set(ii)
ClIntΩ,Θ=ClΩ,Θ(iii)
There exists anξ-infra soft open set Ψ,Θ such that Ψ,Θ⊆~Ω,Θ⊆~ClΨ,ΘProof.
i⇔ii: Let Ω,Θ be an infra soft semiopen set. Then, Ω,Θ⊆~ClIntΩ,Θ. Therefore, ClΩ,Θ⊆~ClIntΩ,Θ. It is well known that ClIntΩ,Θ⊆~ClΩ,Θ. Thus, ClIntΩ,Θ=ClΩ,Θ. The direction ii⇒i is obvious.
ii⇔iii: Since Ω,Θ is an infra soft semiopen set, Ω,Θ⊆~ClIntΩ,Θ. Taking Ψ,Θ=IntΩ,Θ, we obtain Ψ,Θ⊆~Ω,Θ⊆~ClΨ,Θ. Since IntΨ,Θ=IntIntΩ,Θ=IntΩ,Θ=Ψ,Θ, Ψ,Θ is an ξ-infra soft open set. Conversely, let Ψ,Θ is an ξ-infra soft open set such that Ψ,Θ⊆~Ω,Θ⊆~ClΨ,Θ. Then, ClΨ,Θ⊆~ClIntΩ,Θ. By assumption, Ω,Θ⊆~ClIntΩ,Θ which means that Ω,Θ is an infra soft semiopen set.☐Proposition 22.
LetΩ,Θ be a subset of an ISTS T,ξ,Θ. Then, we have the following equivalent properties:
(i)
Ω,Θ is an infra soft semiclosed set(ii)
IntClΩ,Θ⊆~Ω,Θ(iii)
IntClΩ,Θ=IntΩ,Θ(iv)
There exists anξ-infra soft closed set Ψ,Θ such that IntΨ,Θ⊆~Ω,Θ⊆~Ψ,ΘProof.
One can prove it following similar arguments given in the proof of Proposition21.☐Proposition 23.
The class of infra soft semiopen sets is closed under arbitrary unions.Proof.
ConsiderΩj,Θ:j∈J as a family of infra soft semiopen sets. If the index J is an empty set, then ∪~j∈∅Ωj,Θ=Φ which is an infra soft semiopen set. Suppose that J≠∅. Then Ωj,Θ⊆~ClIntΩj,Θ for each j∈J. Consequentially, ∪~j∈JΩj,Θ⊆~∪~j∈JClIntΩj,Θ⊆~ClInt∪~j∈JΩj,Θ. Hence, ∪~j∈JΩj,Θ is infra soft semiopen.☐Corollary 24.
The class of infra soft semiclosed sets is closed under arbitrary intersections.Corollary 25.
The class of infra soft semiopen subsets of an ISTST,ξ,Θ forms a supra soft topology over T.To illustrate that the class of infra soft semiopen sets does not form an infra soft topology, we present the following example.Example 1.
LetT=t1,t2,t3,t4 and Θ=θ1,θ2. Then, ξ=Φ,T~,Ω1,Θ,Ω2,Θ is an infra soft topology on T with Θ as a set of parameters, where
(5)Ω1,Θ=θ1,t1,θ2,t1,
and(6)Ω2,Θ=θ1,t2,θ2,t2.
LetΩ3,Θ=θ1,t1,t3,θ2,t1,t3 and Ω4,Θ=θ1,t2,t3,θ2,t2,t3. Then, Ω3,Θ and Ω4,Θ are infra soft semiopen sets because ClIntΩ3,Θ=θ1,t1,t3,t4,θ2,t1,t3,t4 and ClIntΩ4,Θ=θ1,t2,t3,t4,θ2,t2,t3,t4. But Ω3,Θ∩~Ω4,Θ is not infra soft semiopen because ClIntΩ3,Θ∩~Ω4,Θ=Φ.Proposition 26.
The intersection of infra soft open and infra soft semiopen sets is an infra soft semiopen set.Proof.
LetΩ1,Θ be an infra soft open set and Ω2,Θ be an infra soft semiopen set. Then, Ω1,Θ∩~Ω2,Θ⊆~Ω1,Θ∩~ClIntΩ2,Θ; by Proposition 15, we obtain Ω1,Θ∩~Ω2,Θ⊆~ClΩ1,Θ∩~IntΩ2,Θ=ClIntΩ1,Θ∩~IntΩ2,Θ=ClIntΩ1,Θ∩~Ω2,Θ. Hence, we obtain the desired result.☐Corollary 27.
The union of infra soft closed and infra soft semiclosed sets is an infra soft semiclosed set.Definition 28.
An ISTST,ξ,Θ is said to be infra soft hyperconnected if the intersection of any two nonnull ξ-infra soft open sets is nonnull. Otherwise, T,ξ,Θ is said to be infra soft dishyperconnected.Proposition 29.
The intersection of two infra soft semiopen subsets of an infra soft hyperconnected space is an infra soft semiopen set.Proof.
LetΩ1,Θ and Ω2,Θ be infra soft semiopen sets. If one of them is the null soft set, then, we obtain the desired result. Suppose that Ω1,Θ and Ω2,Θ are nonnull soft sets. According to Proposition 21, there are two ξ-infra soft open sets Ψ1,Θ≠Φ and Ψ2,Θ≠Φ such that Ψ1,Θ⊆~Ω1,Θ⊆~ClΨ1,Θ and Ψ2,Θ⊆~Ω2,Θ⊆~ClΨ2,Θ. By hypothesis of infra soft hyperconnectedness, Ψ1,Θ∩~Ψ2,Θ is a nonnull ξ-infra soft open set. Now, Ψ1,Θ∩~Ψ2,Θ⊆~Ω1,Θ∩~Ω2,Θ⊆~ClΨ1,Θ∩~Ψ2,Θ. Hence, Ω1,Θ∩~Ω2,Θ is an infra soft semiopen set.☐Lemma 30.
LetEτ:T1,ξ1,Θ1⟶T2,ξ2,Θ2 be an infra soft homeomorphism map. Then, for any subset Ω,Θ1, we have the next two results.
(i)
EτIntΩ,Θ1=IntEτΩ,Θ1(ii)
EτClΩ,Θ1=ClEτΩ,Θ1Proof.
To prove (i), letδθ′s∈EτIntΩ,Θ1. Then, there is δθt∈IntΩ,Θ1 such that Eτδθt=δθ′s. This means there exists an infra soft open set Ψ,Θ1 such that δθt∈Ψ,Θ1⊆~Ω,Θ1. Therefore, δθ′s=Eτδθt∈EτΨ,Θ1⊆~EτΩ,Θ1. This implies that δθ′s∈IntEτΩ,Θ1. Thus, EτIntΩ,Θ1⊆~IntEτΩ,Θ1. Conversely, let δθ′s∈IntEτΩ,Θ1. Then, there exists an infra soft open set Ψ,Θ2 such that δθ′s∈Ψ,Θ2⊆~EτΩ,Θ1. Therefore, Eτ−1δθ′s∈Eτ−1Ψ,Θ2⊆~Ω,Θ1. Automatically, we obtain Eτ−1δθ′s∈IntΩ,Θ1. So that, δθ′s∈EτIntΩ,Θ1. Thus, IntEτΩ,Θ1⊆~EτIntΩ,Θ1. Hence, the proof is complete.
Following similar arguments, one can prove (ii).☐Proposition 31.
The infra soft homeomorphism image of an infra soft semiopen set is an infra soft semiopen set.Proof.
ConsiderEτ:T1,ξ1,Θ1⟶T2,ξ2,Θ2 as an infra soft continuous mapping and let Ω,Θ1 be an infra soft semiopen subset of T1,ξ1,Θ1. Then, EτΩ,Θ1⊆~EτClIntΩ,Θ1. It follows from the above lemma that EτΩ,Θ1⊆~ClIntEτΩ,Θ1. Hence, EτΩ,Θ1 is an infra soft semiopen subset of T2,ξ2,Θ2, as required.☐Lemma 32.
ConsiderΩ1,Θ1 and Ω2,Θ2 as subsets of T1,ξ1,Θ1 and T2,ξ2,Θ2, respectively. Then
(i)
ClΩ1,Θ1×Ω2,Θ2=ClΩ1,Θ1×ClΩ2,Θ2(ii)
IntΩ1,Θ1×Ω2,Θ2=IntΩ1,Θ1×IntΩ2,Θ2Proof.
i: Let δθ,ϑt,s∉ClΩ1,Θ1×Ω2,Θ2. Then, there is an infra soft open subset Ψ1,Θ1×Ψ2,Θ2 of T1~×T2~ containing δθ,ϑt,s such that Ω1,Θ1×Ω2,Θ2∩~Ψ1,Θ1×Ψ2,Θ2=ΦΘ1×Θ2. This implies that Ω1,Θ1∩~Ψ1,Θ1=ΦΘ1 or Ω2,Θ2∩~Ψ2,Θ2=ΦΘ2. Therefore, δθt∉ClΩ1,Θ1 or δϑs∉ClΩ2,Θ2. Thus, δθ,ϑt,s∉ClΩ1,Θ1×ClΩ2,Θ2. Hence, ClΩ1,Θ1×ClΩ2,Θ2⊆~ClΩ1,Θ1×Ω2,Θ2. Conversely, let δθ,ϑt,s∉ClΩ1,Θ1×ClΩ2,Θ2. Then, δθt∉ClΩ1,Θ1 or δϑs∉ClΩ2,Θ2. Suppose, without loss of generality, that δθt∉ClΩ1,Θ1. Then, there is an infra soft open subset Ψ1,Θ1 of T1,ξ1,Θ1 containing δθt such that Ω1,Θ1∩~Ψ1,Θ1=ΦΘ1. Obviously, Ψ1,Θ1×T2~ is an infra soft open subset of T1~×T2~ containing δθ,ϑt,s such that Ψ1,Θ1×T2~∩~Ω1,Θ1×Ω2,Θ2=ΦΘ1×Θ2. Therefore, δθ,ϑt,s∉ClΩ1,Θ1×Ω2,Θ2. Thus, ClΩ1,Θ1×Ω2,Θ2⊆~ClΩ1,Θ1×ClΩ2,Θ2. Hence, the proof is complete.
Following similar arguments, one can prove (ii).☐Proposition 33.
The product of infra soft semiopen sets is an infra soft semiopen set.Proof.
LetΩ1,Θ1 and Ω2,Θ2 be infra soft semiopen subsets of T1,ξ1,Θ1 and T2,ξ2,Θ2, respectively. Then, Ω1,Θ1×Ω2,Θ2⊆~ClIntΩ1,Θ1×ClIntΩ2,Θ2. According to the above lemma, we obtain Ω1,Θ1×Ω2,Θ2⊆~ClIntΩ1,Θ1×Ω2,Θ2 which means that Ω1,Θ1×Ω2,Θ2 is an infra soft semiopen subset of T1~×T2~.☐


## 4. Infra Semi-Interior, Infra Semiclosure, Infra Semilimit, and Infra Semiboundary Soft Points of a Soft Set

In this section, we first present the infra soft semi-interior and infra soft semiclosure operators and scrutinize their essential properties. Then, we define infra soft semilimit and infra soft semiboundary soft points of a soft set. We discuss their main features and revel the relationships between them with the help of examples.Definition 34.
LetΩ,Θ be a subset of T,ξ,Θ. Then
(i)
The infra soft semi-interior ofΩ,Θ, denoted by sIntΩ,Θ, is the union of all infra soft semiopen sets that are contained in Ω,Θ(ii)
The infra soft semiclosure ofΩ,Θ, denoted by sClΩ,Θ, is the intersection of all infra soft semiclosed sets containing Ω,ΘProposition 35.
We have the following properties.(i)
Ω,Θ is an infra soft semiopen subset of T,ξ,Θ iff sIntΩ,Θ=Ω,Θ(ii)
Ω,Θ is an infra soft semiclosed subset of T,ξ,Θ iff sClΩ,Θ=Ω,ΘProof.
It follows from Proposition23 and Corollary 24.☐It should be noted that the infra soft open and infra soft closed sets do not satisfy the above two properties.Proposition 36.
LetΩ,Θ be a subset of T,ξ,Θ.
(i)
δθt∈sIntΩ,Θ iff there is an infra soft semiopen set Ψ,Θ such that δθt∈Ψ,Θ⊆~Ω,Θ(ii)
δθt∈sClΩ,Θ iff the intersection of any infra soft semiopen set Ψ,Θ containing δθt and Ω,Θ is nonnullProof.
The proof of (i) is obvious, so we prove (ii).
Letδθt∈sClΩ,Θ. Then, every infra soft semiclosed set contains Ω,Θ contains δθt as well. Suppose that there exists an infra soft semiopen set Ψ,Θ containing δθt such that Ω,Θ∩~Ψ,Θ=Φ. Therefore, Ω,Θ⊆~Ψc,Θ which means that δθt∉sClΩ,Θ. This is a contradiction. Conversely, suppose that there exists an infra soft semiopen set Ψ,Θ containing δθt such that Ω,Θ∩~Ψ,Θ=Φ. Therefore, sClΩ,Θ⊆~Ψc,Θ which means that δθt∉sClΩ,Θ. Hence, we obtain the desired result.☐Proposition 37.
LetΩ,Θ be a subset of T,ξ,Θ. Then,
(i)
sIntΩ,Θc=sClΩc,Θ(ii)
sClΩ,Θc=sIntΩc,ΘProof.
i: sIntΩ,Θc=∪~j∈JΨj,Θ:Ψj,Θ is an infra soft semiopen set contained in Ω,Θc=∩~j∈JΨjc,Θ:Ψjc,Θ is an infra soft semi-closed set containing Ωc,Θ=sClΩc,Θ.
The proof of (ii) is similar to (i).☐Proposition 38.
LetΨ,Θ be an infra soft open set and Λ,Θ be an infra soft closed set in T,ξ,Θ. Then,
(i)
Ψ,Θ∩~sClΩ,Θ⊆~sClΨ,Θ∩~Ω,Θ(ii)
sIntΛ,Θ∪~Ω,Θ⊆~Λ,Θ∪~sIntΩ,ΘProof.i: Let δθt∈Ψ,Θ∩~sClΩ,Θ. Then, δθt∈Ψ,Θ and δθt∈sClΩ,Θ.
This implies thatΓ,Θ∩~Ω,Θ≠Φ for every infra soft semiopen set Γ,Θ containing δθt. It follows from Proposition 26 that Ψ,Θ∩~Γ,Θ is an infra soft semiopen set containing δθt. Therefore, Ψ,Θ∩~Γ,Θ∩~Ω,Θ≠Φ. Now, Γ,Θ∩~Ψ,Θ∩~Ω,Θ≠Φ which means that δθt∈sClΨ,Θ∩~Ω,Θ. Hence, Ψ,Θ∩~sClΩ,Θ⊆~sClΨ,Θ∩~Ω,Θ.
One can prove (ii) following similar arguments.☐Theorem 39.
LetΩ,Θ and Ψ,Θ be subsets of T,ξ,Θ. Then, we have the following properties.
(i)
sIntT~=T~(ii)
sIntΩ,Θ⊆~Ω,Θ(iii)
IfΨ,Θ⊆~Ω,Θ, then sIntΨ,Θ⊆~sIntΩ,Θ(iv)
sIntsIntΩ,Θ=sIntΩ,Θ(v)
sIntΨ,Θ∩~sIntΩ,Θ⊆~sIntΨ,Θ∩~Ω,ΘProof.
i: Since T~ is infra soft semiopen, sIntT~=T~.
(ii) and (iii) are obvious.
(iv): It is clear thatsIntsIntΩ,Θ is the largest infra soft semiopen set contained in sIntΩ,Θ; however, sIntΩ,Θ is an infra soft semiopen set; hence, sIntsIntΩ,Θ=sIntΩ,Θ.
(v): It comes from (iii).☐Theorem 40.
LetΩ,Θ and Ψ,Θ be subsets of T,ξ,Θ. Then, we have the following properties.
(i)
sClΦ=Φ(ii)
Ω,Θ⊆~sClΩ,Θ(iii)
IfΨ,Θ⊆~Ω,Θ, then sClΨ,Θ⊆~sClΩ,Θ(iv)
sClsClΩ,Θ⊆~sClΩ,Θ(v)
sClΨ,Θ∪~Ω,Θ=sClΨ,Θ∪~sClΩ,ΘProof.
It can be proved following similar arguments given in the proof of Theorem39.☐The next example shows that the inclusion relations given in the above two theorems are proper.Example 2.
LetT=t1,t2 and Θ=θ1,θ2. Then, ξ=Φ,T~,Ωj,Θ:j=1,2,3 is an infra soft topology on T over T with Θ as a set of parameters, where
(7)Ω1,Θ=θ1,t1,θ2,∅;(8)Ω2,Θ=θ1,∅,θ2,t1,
and(9)Ω3,Θ=θ1,T,θ2,t2.
LetΨ1,Θ=θ1,t2,θ2,t1. Then, sIntΨ1,Θ=θ1,∅,θ2,t1⊂~Ψ1,Θ and sClΨ1,Θ=θ1,t2,θ2,T⊃~Ψ1,Θ. Also, consider Ψ2,Θ=θ1,t2,θ2,∅. Then, sClΨ1,Θ∪~Ψ2,Θ=θ1,t2,θ2,T⊇~sClΨ1,Θ∩~sClΨ2,Θ=θ1,t2,θ2,t2.Definition 41.
A soft pointδθt is said to be an infra soft semilimit point of a subset Ω,Θ of T,ξ,Θ provided that Ψ,Θ\δθt∩~Ω,Θ≠Φ for every infra soft semiopen set Ψ,Θ containing δθt.The soft set of all infra soft semilimit points ofΩ,Θ is said to be an infra semiderived soft set. It is denoted by Ω,Θis′.Proposition 42.
ConsiderΨ,Θ and Ω,Θ as subsets of T,ξ,Θ. Then,
(i)
Φis′=Φ and T~is′⊆~T~(ii)
IfΨ,Θ⊆~Ω,Θ, then Ψ,Θis′⊆~Ω,Θis′(iii)
Ifδθt∈Ω,Θis′, then δθt∈Ω,Θ\δθtis′(iv)
Ψ,Θis′∪~Ω,Θis′⊆~Ψ,Θ∪~Ω,Θis′.Proof.
Straightforward.☐Theorem 43.
LetΩ,Θ be a subset of T,ξ,Θ. Then,
(i)
IfΩ,Θ is an infra soft semiclosed set, then, Ω,Θis′⊆Ω,Θ(ii)
Ω,Θ∪~Ω,Θis′is′⊆~Ω,Θ∪~Ω,Θis′(iii)
sClΩ,Θ=Ω,Θ∪~Ω,Θis′Proof.
(i)
ConsiderΩ,Θ as an infra soft semiclosed set such that δθt∉Ω,Θ. Then, δθt∈Ωc,Θ. Now, Ωc,Θ is an infra soft semiopen set such that Ωc,Θ∪~Ω,Θ=Φ which means that δθt∉Ω,Θis′. Thus, Ω,Θis′⊆~Ω,Θ(ii)
Considerδθt∉Ω,Θ∪~Ω,Θis′. Then, δθt∉Ω,Θ and δθt∉Ω,Θis′. Therefore, there exists an infra soft semiopen set Ψ,Θ such that(10)Ψ,Θ⋂~Ω,Θ=Φ,
This implies that(11)Ψ,Θ⋂~Ω,Θis′=Φ.
It follows from (10) and (11) that Ψ,Θ∩~Ω,Θ∪~Ω,Θis′=Φ. Thus, δθt∉Ω,Θ∪~Ω,Θis′is′. Hence, Ω,Θ∪~Ω,Θis′is′⊆~Ω,Θ∪~Ω,Θis′, as required.
(iii)
It is clear thatΩ,Θ∪~Ω,Θis′⊆~sClΩ,Θ. Conversely, let δθt∈sClΩ,Θ. Then, for every infra soft semiopen set containing δθt, we have Ω,Θ∩~Ψ,Θ≠Φ. Without loss of generality, let δθt∉Ω,Θ. Then, Ω,Θ\δθt∩~Ψ,Θ≠Φ. Consequentially, δθt∈Ω,Θis′. Hence, the proof is complete.Definition 44.
The infra soft semiboundary points of a subsetΩ,Θ of T,ξ,Θ, denoted by sBΩ,Θ, are all the soft points which belong to the complement of sIntΩ,Θ∪~sIntΩc,Θ.Proposition 45.
LetΩ,Θ be a subset of T,ξ,Θ. Then,
(i)
sBΩ,Θ=sClΩ,Θ⋂~sClΩc,Θ(ii)
sBΩ,Θ=sClΩ,Θ\sIntΩ,ΘProof.
(i)
sBΩ,Θ=δθt∈T~:δθt∉sIntΩ,Θandδθt∉sIntΩc,Θ=δθt∈T~:δθt∉sClΩc,Θcandδθt∉sClΩ,Θc=δθt∈T~:δθt∈sClΩc,Θandδθt∈sClΩ,Θ1.9cm=sClΩ,Θ∩~sClΩc,Θ(ii)
sBΩ,Θ=sClΩ,Θ∩~sClΩc,Θ1.7cm=sClΩ,Θ∩~sIntΩ,Θc1.7cm=sClΩ,Θ\sIntΩ,Θ☐Corollary 46.
LetΩ,Θ be a subset of T,ξ,Θ. Then
(i)
sBΩ,Θ=sBΩc,Θ(ii)
sClΩ,Θ=sIntΩ,Θ∪~sBΩ,ΘProposition 47.
LetΩ,Θ be a subset of T,ξ,Θ. Then,
(i)
Ω,Θ is infra soft semiopen iff sBΩ,Θ∩~Ω,Θ=Φ(ii)
Ω,Θ is infra soft semiclosed iff sBΩ,Θ⊆~Ω,ΘProof.
(i)
sBΩ,Θ∩Ω,Θ=sBΩ,Θ∩sIntΩ,Θ=Φ. Conversely, let δθt∈Ω,Θ. Then δθt∈sIntΩ,Θ or δθt∈sBΩ,Θ. Since sBΩ,Θ∩Ω,Θ=Φ, δθt∈sIntΩ,Θ. Thus, Ω,Θ⊆sIntΩ,Θ which means that Ω,Θ=sIntΩ,Θ. Hence, Ω,Θ is infra soft semiopen(ii)
Ω,Θ is infra soft semiclosed ⇔Ωc,Θ is infra soft semiopen ⇔sBΩc,Θ∩Ωc,Θ=Φ⇔sBΩ,Θ∩Ωc,Θ=Φ⇔sBΩ,Θ⊆Ω,ΘCorollary 48.
A subsetΩ,Θ of T,ξ,Θ is infra soft semiopen and infra soft semiclosed iff sBΩ,Θ=Φ.


## 5. Infra Soft Semihomeomorphism Maps

This section introduces the concepts of infra soft semicontinuous, infra soft semiopen, infra soft semiclosed, and infra soft semihomeomorphism maps. We give some characterizations of each one of these concepts and demonstrate some interrelations between them. Finally, we study the concept of fixed soft points with respect to infra soft semiopen sets.Definition 49.
A soft mappingEτ:T,ξ,Θ⟶S,π,Δ is said to be infra soft semicontinuous at δθt∈T~ if for any infra soft semiopen set Ψ,Δ containing Eτδθt, there is an infra soft semiopen set Ω,Θ containing δθt such that EτΩ,Θ⊆~Ψ,Δ.IfEτ is infra soft semicontinuous at all soft points of the domain, then, it is called infra soft semicontinuous.Theorem 50.
Eτ:T,ξ,Θ⟶S,π,Δ is an infra soft semicontinuous mapping iff the preimage of each infra soft semiopen set is infra soft semiopen.Proof.
Necessity: letΩ,Δ be an infra soft semiopen subset of S,π,Δ. If Eτ−1Ω,Δ=Φ, then, the proof is trivial. So, consider Eτ−1Ω,Δ≠Φ. Now, for any δθt∈Eτ−1Ω,Δ, there, is an infra soft semiopen subset Ψi,Θ of T,ξ,Θ containing δθt such that EτΨi,Θ⊆~Ω,Δ. Thus, δθt∈Ψi,Θ⊆~Eτ−1Ω,Δ and ⋃~Ψi,Θ=Eτ−1Ω,Δ. This means that Eτ−1Ω,Δ is infra soft semiopen.
Sufficiency: letδθt∈T~ and Ψ,Θ be an infra soft semiopen set containing Eτδθt. Then, Eτ−1Ψ,Θ is an infra soft semiopen set containing δθt such that EτEτ−1Ψ,Θ⊆~Ψ,Θ. This means that Eτ is infra soft semicontinuous at δθt. This ends the proof that Eτ is infra soft semicontinuous.☐Example 3.
ConsiderT is the set of real numbers, S is the set of natural numbers and Θ=θ1,θ2. Let Eτ:T,ξ,Θ⟶T,π,Θ and Fν:S,μ,Θ⟶S,π,Θ be two soft mapping such that E,F,τ, and ν are identity mappings, and π is the discrete soft topology (it is also infra soft topology). Let ξ=T,Ω,Θ⊆~T:Ω,Θisfinite and μ=Φ,S,θ1,2,θ2,3,θ1,3,θ2,2 are two infra soft topologies on T and S, respectively. It is clear that every subset of T,ξ,Θ is infra soft semiopen. So that, Eτ is infra soft semicontinuous. On the other hand, Fν is not infra soft semicontinuous because Ψ,Θ=θ1,1,θ2,1 is an infra soft semiopen subset of S,π,Θ, whereas its preimage under a soft mapping Fν is not an infra soft semiopen subset of S,μ,Θ.Theorem 51.
LetEτ:T,ξ,Θ⟶S,π,Δ be an infra soft semicontinuous mapping. Then, we have the following five equivalent statements:
(i)
Eτ is an infra soft semicontinuous mapping(ii)
The preimage of each infra soft semiclosed set is infra soft semiclosed(iii)
sClEτ−1Ω,Δ⊆~Eτ−1sClΩ,Δ for each Ω,Δ⊆~S~(iv)
EτsClΨ,Θ⊆~sClEτΨ,Θ for each Ψ,Θ⊆~T~(v)
Eτ−1sIntΩ,Δ⊆~sIntEτ−1Ω,Δ for each Ω,Δ⊆~S~Proof.
i⇒ii: Let Ω,Δ be an infra soft semiclosed set in S,π,Δ.
Then,Eτ−1Ωc,Δ is an infrasoft semiopen subset of T~. Obviously, Eτ−1Ωc,Δ=T~−Eτ−1Ω,Δ; hence, Eτ−1Ω,Δ is an infra soft semiclosed subset of T~.
ii⇒iii: According to (ii), Eτ−1sClΩ,Δ is an infra soft semiclosed subset of T~. Then, sClEτ−1Ω,Δ⊆~sClEτ−1sClΩ,Δ=Eτ−1sClΩ,Δ.
iii⇒vi: According to (iii), sClEτ−1EτΨ,Θ⊆~Eτ−1sClEτΨ,Θ. Then, EτsClΨ,Θ⊆~EτEτ−1sClEτΨ,Θ⊆~sClEτΨ,Θ.
iv⇒v: According to (iv), EτsClT~−Eτ−1Ω,Δ⊆~sClEτT~−Eτ−1Ω,Δ. Therefore, EτT~−sIntEτ−1Ω,Δ=EτsClT~−Eτ−1Ω,Δ⊆sClS~−Ω,Δ=S~−sIntΩ,Δ. Thus, T~−sIntEτ−1Ω,Δ⊆~Eτ−1S~−sIntΩ,Δ=Eτ−1S~−Eτ−1sIntΩ,Δ. Hence, Eτ−1sIntΩ,Δ⊆~sIntEτ−1Ω,Δ.
v⇒i: Let Ω,Δ be an infra soft open subset of S~. According to (v), Eτ−1Ω,Δ⊆~sIntEτ−1Ω,Δ. This implies that Eτ−1Ω,Δ=sIntEτ−1Ω,Δ. Hence, Eτ is infra soft semicontinuous.☐Theorem 52.
IfEτ:T,ξ,Θ⟶S,π,Δ is infra soft semicontinuous, then, the restriction soft mapping Eτ∣M:M,ξM,Θ⟶S,π,Δ is infra soft semicontinuous provided that M~ is an infra soft open set.Proof.
ConsiderΩ,Δ is an infra soft semiopen set in S,π,Δ. By hypothesis, Eτ−1Ω,Δ is infra soft semiopen. Now, Eτ∣M−1Ω,Δ=Eτ−1Ω,Δ∩~M~. Since M~ is an infra soft open set, it follows from Proposition 26 that Eτ∣M−1Ω,Δ is infra soft semiopen. Hence, Eτ∣M is an infra soft semicontinuous map.☐Proposition 53.
LetEτ:T,ξ,Θ⟶S,π,Δ and Fν:S,π,Δ⟶V,σ,Γ be infra soft semicontinuous. Then, Fν∘Eτ is infra soft semicontinuous.Proof.
Straightforward.☐Definition 54.
A soft mappingEτ:T,ξ,Θ⟶S,π,Δ is said to be infra soft semiopen (resp., infra soft semiclosed) if the image of each infra soft semiopen (resp., infra soft semiclosed) set is infra soft semiopen (resp., infra soft semiclosed).Proposition 55.
Eτ:T,ξ,Θ⟶S,π,Δ is an infra soft semiopen mapping iff EτsIntΩ,Θ⊆~sIntEτΩ,Θ for each subset of Ω,Θ of T~.Proof.
⇒: Let Ω,Θ be a subset of T~. Now, EτsIntΩ,Θ⊆~EτΩ,Θ and sIntΩ,Θ are an infra soft semiopen set. By hypothesis, EτsIntΩ,Θ is infra soft semiopen. Therefore, EτsIntΩ,Θ⊆~sIntEτΩ,Θ.
⇐: Let Λ,Θ be an infra soft open subset of T~. Then, EτΩ,Θ⊆~sIntEτΩ,Θ. Therefore, EτΩ,Θ=sIntEτΩ,Θ which means that Eτ is an infra soft semiopen map.☐Proposition 56.
Eτ:T,ξ,Θ⟶S,π,Δ is an infra soft semiclosed mapping iff sClEτΩ,Θ⊆~EτsClΩ,Θ for each subset Ω,Θ of T~.Proof.
⇒: Let Eτ be an infra soft semiclosed mapping and Ω,Θ be a subset of T~. By hypothesis, EτsClΩ,Θ is infra soft semiclosed. Since EτΩ,Θ⊆~EτsClΩ,Θ, sClEτΩ,Θ⊆~EτsClΩ,Θ.
⇐: Suppose that Ω,Θ is an infra soft semiclosed subset of T~. By hypothesis, EτΩ,Θ⊆~sClEτΩ,Θ⊆~EτsClΩ,Θ=EτΩ,Θ. Therefore, EτΩ,Θ is infra soft semiclosed. Hence, Eτ is an infra soft semiclosed map.☐Proposition 57.
The concepts of infra soft semiopen and infra soft semiclosed mappings are equivalent under bijectiveness.Proof.
It comes from the fact that a bijective soft mappingEτ:T,ξ,Θ⟶S,π,Δ implies that EτΩc,Θ=EτΩ,Θc.☐Proposition 58.
LetEτ:T,ξ,Θ⟶S,π,Δ and Fν:S,π,Δ⟶V,σ,Γ be two soft maps. Then,
(i)
IfEτ and Fν are infra soft semiopen maps, then, Fν∘Eτ is an infra soft semiopen map(ii)
IfFν∘Eτ is an infra soft semiopen mapping and Eτ is a surjective infra soft semicontinuous map, then, Fν is an infra soft semiopen map(iii)
IfFν∘Eτ is an infra soft semiopen mapping and Fν is an injective infra soft semicontinuous map, then, Eτ is an infra soft semiopen mapProof.
(i)
Straightforward(ii)
ConsiderΩ,Δ as an infra soft semiopen subset of S,π,Δ. By hypothesis, Eτ−1Ω,Δ is an infra soft semiopen subset of T,ξ,Θ. Again, by hypothesis, Fν∘EτEτ−1Ω,Δ is an infra soft semiopen subset of V,σ,Γ. Since Eτ is surjective, then, Fν∘EτEτ−1Ω,Δ=FνEτEτ−1Ω,Δ=FνΩ,Δ. Hence, Fν is an infra soft semiopen map(iii)
ConsiderΩ,Θ as an infra soft semiopen subset of T,ξ,Θ. By hypothesis, Fν∘EτΩ,Θ is an infra soft semiopen subset of V,σ,Γ. Again, by hypothesis, Fν−1Fν∘EτΩ,Θ is an infra soft semiopen subset of S,π,Δ. Since Fν is injective, then, Fν−1Fν∘EτΩ,Θ=Fν−1FνEτΩ,Θ=EτΩ,Θ. Hence, Eτ is an infra soft semiopen map.☐The following result can be proved following similar arguments given in proposition’ proof above.Proposition 59.
LetEτ:T,ξ,Θ⟶S,π,Δ and Fν:S,π,Δ⟶V,σ,Γ be two infra soft maps. Then, the following statements hold.
(i)
IfEτ and Fν are infra soft semiclosed maps, then, Fν∘Eτ is an infra soft semiclosed map(ii)
IfFν∘Eτ is an infra soft semiclosed mapping and Eτ is a surjective infra soft semicontinuous map, then, Fν is an infra soft semiclosed map(iii)
IfFν∘Eτ is an infra soft semiclosed mapping and Fν is an injective infra soft semicontinuous map, then, Eτ is an infra soft semiclosed mapDefinition 60.
A bijective soft mappingEτ:T,ξ,Θ⟶S,π,Δ is said to be an infra soft semihomeomorphism if it is infra soft semicontinuous and infra soft semiopen.We cancel the proofs of the next two results because they are easy.Proposition 61.
LetEτ:T,ξ,Θ⟶S,π,Δ and Fν:S,π,Δ⟶V,σ,Γ be infra soft semihomeomorphism maps. Then, Fν∘Eτ is an infra soft semihomeomorphism map.Proposition 62.
IfEτ:T,ξ,Θ⟶S,π,Δ is a bijective soft map, then, the following statements are equivalent.
(i)
Eτ is an infra soft semihomeomorphism(ii)
Eτ and Eτ−1 is infra soft semicontinuous(iii)
Eτ is infra soft semiclosed and infra soft semicontinuousProposition 63.
IfEτ:T,ξ,Θ⟶S,π,Δ is an infra soft semihomeomorphism map, then, the following statements hold for each Ω,Θ∈SXA.
(i)
EτsIntΩ,Θ=sIntEτΩ,Θ(ii)
EτsClΩ,Θ=sClEτΩ,ΘProof.
i: According to Proposition 55 (i), we obtain EτsIntΩ,Θ⊆~sIntEτΩ,Θ.
Conversely, letδκs∈sIntEτΩ,Θ. Then, there is an infra soft semiopen set Ψ,Δ such that δκs∈Ψ,Δ⊆~EτΩ,Θ. By hypothesis, δθt=Eτ−1δκs∈Eτ−1Ψ,Δ⊆~Ω,Θ such that Eτ−1Ψ,Δ is an infra soft semiopen set. So that, δθt∈sIntΩ,Θ which means that δκs∈EτsIntΩ,Θ.
One can achieve item (ii) following similar arguments.☐Theorem 64.
The property of an infra soft semidense set is an infra soft topological invariant.Proof.
LetEτ:T,ξ,Θ⟶S,π,Δ be an infra soft semihomeomorphism mapping and consider Ω,Θ as an infra soft semidense subset of T,ξ,Θ, i.e., sClΩ,Θ=T~. It comes from Proposition 63 (ii) that sClEτΩ,Θ=EτsClΩ,Θ=EτT~=sClS~=S~. Thus, EτΩ,Θ is an infra soft semidense set in S,π,Δ, as required.☐We complete this section by studying the concept of fixed soft points with respect to infra soft semiopen sets. For more details in fixed soft points in the crisp setting, see [27–29].Definition 65.
We say thatT,ξ,Θ has a semifixed soft point property provided that for every infra soft semicontinuous mapping Eτ:T,ξ,Θ⟶T,ξ,Θ, there exists δθs∈T such that Eτδθs=δθs.Proposition 66.
The property of being a semifixed soft point is preserved under an infra soft semihomeomorphism.Proof.
ConsiderT1,ξ1,Θ1 and T2,ξ2,Θ2 as two infra soft semihomeomorphism. This means that there exists a bijective soft mapping Eτ:T1,ξ1,Θ1⟶T2,ξ2,Θ2 such that Eτ and Eτ−1 are infra soft semicontinuous. Suppose that T1,ξ1,Θ1 has the property of semifixed soft point. That is any infra soft semicontinuous mapping Eτ:T1,ξ1,Θ1⟶T1,ξ1,Θ1 has a semifixed soft point. Now, consider Cτ:T2,ξ2,Θ2⟶T2,ξ2,Θ2 is infra soft semicontinuous. It is clear that Cτ∘Eτ:T1,ξ1,Θ1⟶T2,ξ2,Θ2 is infra soft semicontinuous. Therefore, Eτ−1∘Cτ∘Eτ:T1,ξ1,Θ1⟶T1,ξ1,Θ1 is infra soft semicontinuous. Since T1,ξ1,Θ1 has a semifixed soft point property, Eτ−1hτEτδθs=δθs for some δθs∈T~. Thus, EτEτ−1hτEτδθs=Eτδθs. This implies that hτEτδθs=Eτδθs. Hence, Eτδθs is a semifixed soft point of Cτ which means that T2,ξ2,Θ2 has a semifixed soft point property.☐


## 6. Concluding Remark and Further Work

This article contributes to the expanding literature on soft topological spaces. The obtained results demonstrate that most soft topological properties of the presented concepts are preserved in structure of infra soft topologies which means we can dispense of some topological stipulations. This gives an advantage of discussing soft topological ideas via infra soft topologies because it relaxes the restrictions imposed in the study. The obtained results in this manuscript and those given in [17–19] validate this viewpoint.On the other hand, there are a few properties of some topological concepts that are partially losing via infra soft topology such as the equivalence between an infra soft semiopen setΩ,Θ and the existence of an infra soft open set Ψ,Θ such that Ψ,Θ⊆~Ω,Θ⊆~ClΨ,Θ. However, we have addressed this matter by defining an ξ-infra soft open set and proving the counterpart equivalence as given in Proposition 21. As we have shown in Corollary 25 that the class of infra soft semiopen subsets on ISTSs forms a new generalization of soft topology called a supra soft topology.This work considers a promising line for future work; for example, we will complete introducing the main topological concepts using infra soft semiopen sets such as soft separation axioms, soft compact, and soft connected spaces. Our roadmap for research also comprises the examination of the concepts and results initiated herein using another generalization of infra soft open sets such as infra softα-open and infra soft b-open sets. Moreover, we will introduce new types of rough approximations using these generalizations of infra soft open sets and apply them to improve the accuracy measures of sets.



---

*Source: 5716876-2021-11-27.xml*