Infra Soft Semiopen Sets and Infra Soft Semicontinuity

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.


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, Çagman et al. [8] and Shabir and Naz [9] made use of soft sets to define soft topological spaces. Whereas, Çagman 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: Section 2 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.

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

Soft Set Theory
Definition 1 (see [1]). Consider Θ as a set of parameters, T a universal set, and 2 T the power set of T . An ordered pair ðΩ, ΘÞ is called a soft set over T provided that Ω : Θ ⟶ 2 T is a crisp mapping. A soft set is expressed as ðΩ, ΘÞ = f ðθ, ΩðθÞÞ: θ ∈ Θ and ΩðθÞ ∈ 2 T g. We call ΩðθÞ a θ-approximate of ðΩ, ΘÞ.
The class of all soft sets over T under a set of parameters Θ is symbolized by CðT Θ Þ.
Definition 3 (see [5]). We call ðΩ, ΘÞ an absolute (resp., a null) soft set over T if the image of each parameter of Θ under a mapping Ω : Θ ⟶ 2 T is the universal set T (resp., empty set).
The absolute and null soft sets are symbolized byT and Φ, respectively.
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 E τ is injective, or surjective according to its two crisp maps E and τ.

Infra Soft Topological Spaces
Definition 12 (see [17]). A family ξ 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 triple ðT , ξ, ΘÞ 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 ðT , ξ, ΘÞ stable if all its infra soft open sets are stable.

Journal of Function Spaces
Definition 13 (see [17] A property is called an infra soft topological property (briefly, IST property) if it is preserved by any infra soft homeomorphism.

Proposition 19.
Let fðT k , ξ k , Θ k Þ: k ∈ Kg be a family of ISTSs. Then, ξ = f Q k∈K ðθ k , Θ k Þ: ðθ k , Θ k Þ ∈ τ k g is an infra soft topology on T = Q k∈K T k under a set of parameters We call ξ given in proposition above, a product of infra soft topologies, and ðT , ξ, BÞ a product of infra soft spaces.

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.
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.
Journal of Function Spaces 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.
The soft set of all infra soft semilimit points of ðΩ, ΘÞ is said to be an infra semiderived soft set. It is denoted by ðΩ, ΘÞ is′ .
This implies that Journal of Function Spaces It follows from (10) and (11)

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.
If E τ is infra soft semicontinuous at all soft points of the domain, then, it is called infra soft semicontinuous.

☐
The following result can be proved following similar arguments given in proposition' proof above.
(i) If E τ and F ν are infra soft semiclosed maps, then, F ν ∘ E τ is an infra soft semiclosed map (ii) If F ν ∘ E τ is an infra soft semiclosed mapping and E τ is a surjective infra soft semicontinuous map, then, F ν is an infra soft semiclosed map (iii) If F ν ∘ E τ is an infra soft semiclosed mapping and F ν is an injective infra soft semicontinuous map, then, E τ is an infra soft semiclosed map Definition 60. A bijective soft mapping E τ : ðT , ξ, ΘÞ ⟶ ð S, π, ΔÞ 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 66. The property of being a semifixed soft point is preserved under an infra soft semihomeomorphism.
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][18][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 set ðΩ, ΘÞ and the existence of an infra soft open set ðΨ, ΘÞ such that ðΨ,ΘÞ ⊆~ðΩ,ΘÞ ⊆~ClðΨ, ΘÞ. 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 soft α-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.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare no conflicts of interest.