Hindawi Journal of Applied Mathematics Volume 2020, Article ID 1746103, 11 pages https://doi.org/10.1155/2020/1746103

Research Article Soft Separation Axioms and Fixed Soft Points Using Soft Semiopen Sets

T. M. Al-shami

Department of Mathematics, Sana'a University, Sana'a, Yemen

Correspondence should be addressed to T. M. Al-shami; [email protected]

Received 1 August 2020; Revised 28 October 2020; Accepted 30 October 2020; Published 16 November 2020

Academic Editor: Bruno Carpentieri

Copyright © 2020 T. M. Al-shami. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The importance of soft separation axioms comes from their vital role in classifications of soft spaces, and their interesting properties are studied. This article is devoted to introducing the concepts of tt-soft semi-Tiði =0,1,2,3,4Þ and tt-soft semiregular spaces with respect to ordinary points. We formulate them by utilizing the relations of total belong and total nonbelong. The advantages behind using these relations are, first, generalization of existing comparable properties on general and, second, eliminating the stability shape of soft open and closed subsets of soft semiregular spaces. By some examples, we show the relationships between them as well as with soft semi-Tiði =0,1,2,3,4Þ and soft semiregular spaces. Also, we explore under what conditions they are kept between soft topology and its parametric . We characterize a tt-soft semiregular space and demonstrate that it guarantees the equivalence of tt-soft semi-Tiði =0,1,2Þ. Further, we investigate some interrelations of them and some soft topological notions such as soft compactness, product soft spaces, and sum of soft topological spaces. Finally, we define a concept of semifixed soft point and study its main properties.

1. Introduction In 2011, Shabir and Naz [6] defined a soft over a family of soft sets. They constructed the In daily life, human beings face different kinds of uncer- fundamental notions of soft topological spaces such as the tainties in fields such as economics, environmental and social operators of soft and , soft subspace, and soft sciences, engineering, medicine etc. To tackle such uncer- separation axioms. Min [7] continued studying soft separa- tainties, different approaches were proposed like fuzzy and tion axioms and corrected some alleged results in [6]. Then, rough set theories. However, they have their own limitations. Aygünoglu and Aygün [8] scrutinized the concept of com- Molodtsov [1] introduced an effective tool that is free from pactness on soft setting. By exploiting one of the diverges these limitations to solve uncertainties, namely soft set. This between soft sets and crisp sets, Hida [9] studied a strong tool does not require specific type of parameters. Instead, it type of soft compactness. Al-shami [10, 11] revised many accommodates all types of parameters such as words, num- results given in many studies of soft separation axioms. Then, bers, sentences, and functions. After that, Maji et al. [2] estab- Al-shami et al. [12] introduced a weak type of soft compact- lished some fundamental operations between two soft sets ness, namely almost soft compactness. The authors of [13] and then tackled one of their applications in decision- presented soft maps by using two crisp maps, one of them making problems [3]. But the authors of [4] demonstrated between the sets of parameters and the second one between some shortcomings of the operations defined in [2]. They the universal sets. However, the authors of [14] introduced also explored novel operations between soft sets which soft maps by using the concept of soft points. Some applica- became a goal of many studies on algebraic soft structures. tions of different types of soft maps were the goal of some Al-shami and El-Shafei [5] defined new types of soft subset articles (see [13–15]). and equality relations and then applied them on soft linear Until 2018, the belong and nonbelong relations that equations. utilized in these studies are those given by [6]. In 2018, the 2 Journal of Applied Mathematics

authors of [16] came up new relations of belong and nonbe- In this study, we use a symbol GP to refer a soft set, and ∈ long between an ordinary points and soft set, namely partial we identify it as ordered pairs GP = fðp, GðpÞÞ: p P and belong and total nonbelong relations. In fact, these relations GðpÞ ∈ 2Xg. widely open the door to study and redefine many soft topo- A class of all soft sets defined over X with P is symbolized logical notions. This leads to obtain many fruitful properties by SðXPÞ. and changes which can be seen significantly on the study of – soft separation axioms as it was shown in [16 18]. fi Das and Samanta [19] studied the concept of a soft metric De nition 2. [33] GP is said to be a subset of HP, symbolized ⊆~ ⊆ ∈ based on the soft real set and soft real numbers given in [20]. by GP HP,ifGðpÞ HðpÞ for each p P. Wardowski [14] tackled the fixed point in the setup of soft ⊆~ ⊆~ topological spaces. Abbas et al. [21] presented soft contrac- We write GP = HP provided that GP HP and HP GP. tion mappings and established a soft Banach fixed point the- orem in the framework of soft metric spaces. Recently, many fi ∈ De nition 3. [6, 16] For a soft set GP and w X, we say that researchers explored fixed point findings in soft metric type – ∈ spaces (see, for example, [22 25]). Some results related to (i) w GP, it is read as follows: w totally belongs to GP, intuitionistic fuzzy soft sets were investigated in [26, 27]. if w ∈ GðpÞ for each p ∈ P Recently, Alcantud [28] has discussed the properties of topo- ∈ logical separability on soft setting. (ii) w/GP, it is read as follows: w does not partially ∈ ∈ One of the significant ideas that helps to prove some belong to GP,ifw/GðpÞ for some p P properties and removes some problems on soft topology is (iii) w ⋐ G , it is read as follows: w partially belongs to fi fi P the concept of a soft point. It was rst de ned by Zorlutuna G ,ifw ∈ GðpÞ for some p ∈ P and Çakir [29] in order to study the interior points of a soft P ⋐ set and soft neighborhood systems. Then, [19, 30] simulta- (iv) w/GP, it is read as follows: w does not totally belong ∈ ∈ neously redefined soft points to discuss soft metric spaces. to GP,ifw/GðpÞ for each p P. In fact, the recent definition of a soft point makes similarity between many set-theoretic properties and their counterparts fi on soft setting. Two types of soft topologies, namely enriched De nition 4. [4] The relative complement of GP is another soft topology and extended soft topology, were studied in c c : ⟶ X fi c soft set GP, where a map G P 2 is de ned by G ðpÞ = [8, 30], respectively. Al-shami and Kocinac [31] discussed X \ GðpÞ for all p ∈ P. the equivalence between these two topologies and obtained interesting results. Recently, they [32] have introduced the fi concept of nearly soft Menger spaces and investigated De nition 5. [2, 16, 19, 30] A soft set ðGPÞ over X is said to be main characteristics. We organized this paper as follows: After this introduc- (i) a null soft set (resp., an absolute soft set), symbolized Φ~ ~ tion, Section 2 addresses the basic principles about the soft by (resp., X), if GðpÞ is the empty (resp., universal) ∈ sets and soft topologies. In Section (, we introduce the con- set for each p P cepts of tt-soft semi-Tiði =0,1,2,3,4Þ and tt-soft semiregu- x ∈ ∈ (ii) a soft point Sp if there are p P and x X so that G lar spaces with respect to ordinary points by using the ′ ∅ ′ ∈ relations of total belong and total nonbelong. The relation- ðpÞ = fxg and Gðp Þ = for each p P \ fpg.We x ∈ ∈ ships between them and their main properties are discussed write that Sp GP if x GðpÞ with the help of interesting examples. In Section 4, we explore (iii) a stable soft set if all p-approximate are equal A and a semifixed soft point theorem and study some main proper- it is symbolized by A~. In particular, we symbolize by ties. In particular, we conclude under what conditions semi- x if A = fxg fixed soft points are preserved between a soft topology and its P parametric topologies. Finally, conclusions of the paper in (iv) a countable (resp., finite) soft set if GðpÞ is countable Section 5. (resp., finite) for each p ∈ P. Otherwise, it is said to be uncountable (resp., infinite). 2. Preliminaries fi To well understand the results obtained in this study, we shall De nition 6. [2, 4] The intersection and union of GP and HP mind some essential concepts, definitions, and properties are defined as follows: from the literature. ∩~ (i) Their intersection, symbolized by GP HP, is a soft : ⟶ X set UP, where a mapping U P 2 is given by 2.1. Soft Sets UðpÞ = GðpÞ ∩ HðpÞ fi X ~∪ De nition 1. [1] A map G of P into the power set 2 of a non- (ii) Their union, symbolized by GP HP, is a soft set UP, X is called a soft set over X, where X is the universal where a mapping U : P ⟶ 2X is given by UðpÞ = set and P is a set of parameters. GðpÞ ∪ HðpÞ. Journal of Applied Mathematics 3

The union and intersection operators were generalized Definition 15. [35] The intersection of all soft semiclosed sets Ω for any number of soft sets in a similar way. in ðX, , PÞ containing GP is the soft preclosure of GP.Itis symbolized by G s. fi P De nition 7. [34] The Cartesian product of GP and H F, symbolized by G × H ,isdefined as ðG × HÞðp, f Þ = ∈  s ff P×F It is clear that x GP i for each soft semiopen set UP GðpÞ × Hðf Þ for each ðp, f Þ ∈ P × F. ∩~ ≠ Φ~ x ∈  s ff totally containing x we have GP UP ; and Sp GP i for each soft semiopen set U totally containing Sx we have Definition 8. [13] A soft mapping of SðX Þ into SðY Þ, sym- P p A B ∩~ ≠ Φ~ bolized by f ϕ,isdefined by two crisp maps f : X ⟶ Y and GP UP . ϕ : A ⟶ B so that the image of G is a set in SðX Þ and A A Definition 16. [36] ðX, Ω, PÞ which does not contain disjoint pre-image of H is a set in SðY Þ and they are defined as B B soft open sets is called soft hyperconnected. follows. The following result will help us establish some proper- (i) f ϕðG Þ = ðf ϕðGÞÞ is a set in SðY Þ, where f ϕðGÞðbÞ S A B B ties of soft semiseparation axioms and soft semicompact = −1 f ðGðaÞÞ for each b ∈ B a∈ϕ ðbÞ spaces (see, for example, Theorem 42 and Proposition 49). −1 −1 It implies that the family of soft semiopen subsets of a soft (ii) f ϕ ðH Þ = ðf ϕ ðHÞÞ is a set in SðX Þ, where B A A hyperconnected space ðX, Ω, PÞ forms a new soft topology −1 −1 ϕ ∈ f ϕ ðHÞðaÞ = f ðHð ðaÞÞÞ for each a A. Ωsemi over X that is finer than Ω.

Theorem 17. [37] A class of soft semiopen subsets of a soft fi : ⟶ λ De nition 9. [29] f ϕ SðXAÞ SðYBÞ is called a soft hyperconnected space forms a soft topology. map provided that ϕ and f are λ maps, where λ ∈ finjective, surjective,andbijectiveg. Proposition 18. Let Y~ be soft in ðX, Ω, PÞ. Then 2.2. Soft Topology (1) If ðH, PÞ is soft semiopen and Y~ is soft open set in ð Ω Þ ð Þ ∩~ ð Þ Definition 10. [6] A family Ω of soft sets over X under a fixed X, , P , then H,P Y, P is a soft semiopen set ð Ω Þ set of parameters P is said to be a soft topology on X if it sat- in Y, Y , P fi is es the following: (2) If Y~ is soft open in ðX, Ω, PÞ and ðH, PÞ is a soft semi- open in ðY, Ω , PÞ, then ðH, PÞ is a soft semiopen set ~ Φ~ Ω Y (i) X and are elements of in ðX, Ω, PÞ. (ii) The intersection of a finite number of Ω is an ele- Ω ment of Definition 19. [35] ðX, Ω, PÞ is said to be (iii) The union of an arbitrary number of Ω is an element of Ω. (i) soft semi-T0 if there is a soft semiopen set UP for ≠ ∈ ∈ ∈ ∈ each x y X satisfying x UP and y/UP;ory ∈ UP and x/UP We will call a term of soft topological space on the triple Ω (ii) soft semi-T1 if there are two soft semiopen sets UP ðX, , PÞ, where a terminology of soft open is given for ≠ ∈ ∈ ∈ Ω and VP for each x y X satisfying x UP and y/ every member in and a terminology of soft closed is ∈ ∈ given for a relative complement of a member of Ω. UP; and y VP and x/VP

(iii) soft semi-T2 if there are two disjoint soft semiopen Proposition 11. Ω Ω ≠ ∈ ∈ [6] In ðX, , PÞ, a family p = fGðpÞ: sets UP and VP for each x y X satisfying x UP ∈ Ω ∈ ∈ GP g is a classical topology on X for each p P. and y VP

Ω (iv) soft semiregular if every soft semiclosed set HP so We will call a term of parametric topology on p and a ∈ Ω Ω that x/HP, contains two disjoint soft semiopen term of parametric topological space on ðX, pÞ. ⊆~ ∈ sets UP and VP satisfying HP UP and x VP fi Ω (v) soft seminormal if we separate every two disjoint soft De nition 12. [6] ðY, Y , PÞ is called a soft subspace of ðX, Ω ∅≠ ⊆ Ω ~ ∩ : ∈ Ω semiclosed sets by two disjoint soft semiopen sets , PÞ, where Y X and Y = fY GP GP g. (vi) soft semi-T (resp., soft semi-T ) if it is both soft fi 3 4 De nition 13. [35] GP is called a soft semiopen subset of ðX, semiregular (resp., soft seminormal) and soft semi- Ω, PÞ if G ⊆ ~ ðint ð ÞÞ. P cl GP T1-space Theorem 14. [35] Every soft open set is soft semiopen, and the union of an arbitrary number of soft semiopen sets is soft Definition 20. [38] A soft semiopen cover of ðX, Ω, PÞ is a ~ ~ semiopen. family of soft semiopen sets fG : i ∈ Ig so that X = ∪ ∈ G . iP i I iP 4 Journal of Applied Mathematics

Definition 21. ðX, Ω, PÞ is said to be Definition 28. [39] ðX, Ω, PÞ given in the above proposition is said to be the sum of soft topological spaces and is symbolized x y ~ (i) soft semi-T′ [37] if for every S ≠ S ∈ X, there are by ð ⊕ ∈ X , Ω, PÞ. 2 p p′ i I i twodisjoint soft semiopen sets U and V containing P P ~ x y Theorem 29. ⊆ ⊕g S and S , respectively [39] A soft set ðG,PÞ i ∈I Xi is soft semio- p p′ ⊕ Ω pen (resp., soft semiclosed) in ð i∈IXi, , PÞ if and only if ∩~ f (ii) soft semicompact [38] if every soft semiopen cover of all ðG,PÞ Xi are soft semiopen (resp., soft semiclosed) in ~ fi Ω X has a nite subcover. ðXi, i, PÞ.

Proposition 30. [14] Let gφ : ðX, Ω, PÞ ⟶ ðX, Ω, PÞ be a Proposition 22. ∩~ n ~ x x [37] A soft semicompact (resp., stable soft soft map so that n∈ℕgφðXÞ is a soft point Sp. Then, Sp is a ′ semicompact) set in a soft semi-T2-space (resp., soft semi-T2 unique fixed point of gφ. -space) is soft semiclosed. Theorem 31. [29] Let Ω = fG × F : G ∈ Ω and F ∈ θg, To study the properties that preserved under soft A B A B ∗ where ðX, Ω, AÞ and ðY, θ, BÞ are two soft topological spaces. semi -homeomorphism maps, the concept of a soft semi- A class of all arbitrary union of members of Ω defines a soft irresolute map will be presented in this work under the topology over X × Y under a fixed set of parameters A × B. name of a soft semi∗-continuous map. Lemma 32. Let ðG, PÞ and ðH, PÞ be two subsets of ðX, Ω, PÞ θ Definition 23. [37] gφ : ðX, Ω, PÞ ⟶ ðY, θ, PÞ is called soft and ðY, , PÞ, respectively. Then semi∗ − continuous if the inverse image of each soft semiopen set is soft semiopen. (i) clðG, PÞ × clðH, PÞ = clððG, PÞ × ðH, PÞÞ (ii) int ðG, PÞ × int ðH, PÞ = int ððG, PÞ × ðH, PÞÞ. Proposition 24. [38] The soft semi∗ − continuous image of a soft semicompact set is soft semicompact. 3. Semisoft Separation Axioms fi The de nitions of soft semicontinuous, soft semiopen, In this section, the concepts of tt-soft semi-Tiði =0,1,2,3,4Þ soft semiclosed, and soft semihomeomorphism maps were and tt-soft semiregular spaces are introduced, where tt denotes given in [35] in a similar way of their counterparts in classical the total belong and total nonbelong relations that are utilized topology. in the definitions of these concepts. The relationships between them are shown, and their main features are studied. In addi- tion, their behaviours with the concepts of hereditary, topolog- Definition 25. A soft topology Ω on X is said to be ical, and additive properties are investigated. Some examples are provided to elucidate the obtained results. (i) an enriched soft topology [8] if all soft sets GP so that GðpÞ = ∅ or X are members of Ω First of all, we see that it is necessary to classify contain- ment into several categories as it is shown in remark below. Ω : ∈ fi fi (ii) an extended soft topology [30] if = fGP GðpÞ Factually, this classi cation will play a vital role in rede ning Ω ∈ Ω p for each p Pg, where p is a parametric topology many soft theoretic-set and soft topological concepts, in par- on X. ticular, the concepts of soft interior and closure operators, soft compactness, and soft separation axioms.

č ∈ Al-shami and Ko inac [31] proved that extended and Remark 33. For a soft set GP and x X, we have the following enriched soft topologies are identical and obtained some use- notes: ful results that help to discover the interrelations between soft ∈ topology and its parametric topologies. (i) GP totally contains x if x GP ∈ (ii) GP does not partially contain x if x/GP Theorem 26. [31] A subset ðF, PÞ of an extended soft topo- ⋐ logical space ðX, Ω, PÞ is soft semiopen if and only if each (iii) GP partially contains x if x GP ⋐ p-approximate element of ðF, PÞ is semiopen. (iv) GP does not totally contain x if x/GP.

Proposition 27. Ω ∈ [39] Let fðXi, i, PÞ:i Ig be a classS of Definition 34. ðX, Ω, PÞ is said to be mutually disjoint soft topological spaces and X = i∈IXi. Ω ⊆~ ~ : ∩~ f Then the class = fðG,PÞ X ðG,PÞ Xi is a soft open set (i) tt-soft semi-T0 if there exists a soft semiopen set UP Ω ∈ : ≠ ∈ ∈ ⋐ in ðXi, i, PÞ for every i Ig for every x y X satisfying x UP and y/UP or ∈ ⋐ forms a soft topology on X with P. y UP and x/UP Journal of Applied Mathematics 5

≠ ∈ (ii) tt-soft semi-T1 if there exist soft semiopen sets UP x y X. Then, there exist two soft semiopen sets UP and VP ≠ ∈ ∈ ∈ ∈ ∈ ∈ and VP for every x y X satisfying x UP and so that x UP and y/UP; and y VP and x/VP. Since UP and ⋐ ∈ ⋐ y/UP; and y VP and x/VP VP are soft semiopen subsets of a soft semiregular space, then ⋐ ⋐ Ω they are stable. So y/UP and x/VP. Thus, ðX, , PÞ is tt-soft (iii) tt-soft semi-T2 if there exist two disjoint soft semio- ≠ ∈ semi-T1. Hence, we obtain the desired result. pen sets UP and VP for every x y X satisfying fi ∈ ⋐ ∈ ⋐ The following examples clari es that the converse of the x UP and y/UP; and y VP and x/VP above proposition is not always true.

(iv) tt-soft semiregular if for every soft semiclosed set HP ⋐ Ω Φ~ ~ : so that x/HP, there exist disjoint soft semiopen sets Example 1. Let P = fp1, p2g. A family = f , X, Gi i =1, ⊆~ ∈ P UP and VP so that HP UP and x VP 2, ⋯,6g is a soft topology on X = fv, wg, where (v) tt-soft semi-T (resp., tt-soft semi-T )ifitis 3 4 G = fgðÞp , fgv , ðÞp , X both tt-soft semiregular (resp., soft seminormal) 1P 1 2 and tt-soft semi-T . 1 G = ðÞp , X , ðÞp , w 2P fg1 2 fg

G = fgðÞp , fgv , ðÞp , fgw Remark 35. It can be noted that if FP and GP are disjoint soft 3P 1 2 ∈ ff ⋐ Ω ð1Þ set, then x FP i x/GP. This implies that ðX, , PÞ is a tt-soft ff G = fgðÞp , fgw , ðÞp ,∅ semi-T2-space i is a soft semi-T2-space. That is, the concepts 4P 1 2 of a tt-soft semi-T2-space and a soft semi-T2-space are G = ðÞp ,∅ , ðÞp , v equivalent. 5P fg1 2 fg

Ω G = fgðÞp , fgw , ðÞp , fgv : We can say that ðX, , PÞ is tt-soft semi-T2 if there exist 6P 1 2 ≠ ∈ two disjoint soft semiopen sets UP and VP for every x y X so that UP and VP totally contain x and y, respectively. Since the soft sets fðp1, fvgÞ, ðp2, fvgÞg and fðp1, fwgÞ, Ω ðp2, fwgÞg are not soft semiopen subsets of ðX, , PÞ, then Ω Remark 36. The soft semiregular spaces imply a strict condi- it is not tt-soft semi-T0. However, it is clear that ðX, , PÞ tion on the shape of soft semiopen and soft semiclosed subsets. is a soft semi-T1-space. Also, it is soft seminormal. Therefore, To explain this matter, let FP be a soft semiclosed set so that it is soft semi-T4. ∈ x/HP. Then, we have two cases: Example 2. Let P = fp , p g. A family Ω = fΦ~, X~, G : i =1, ′ ′ 1 2 iP (i) There are p, p ∈ P so that x/∈HðpÞ and x ∈ Hðp Þ. 2, 3, 4g is a soft topology on X = fv, wg, where This case is impossible because there do not exist two disjoint soft sets UP and VP containing x and G = fgðÞp , fgv , ðÞp , fgv 1P 1 2 HP, respectively ∈ ∈ G = fgðÞp , fgw , ðÞp , fgw (ii) For each p P, x/HðpÞ. This implies that HP must be 2P 1 2 stable. ð2Þ G = ðÞp , v , ðÞp , X 3P fg1 fg 2

As a direct consequence, we infer that every soft semi- G = fgðÞp ,∅ , ðÞp , fgw : 4P 1 2 closed and soft semiopen subsets of a soft semiregular space must be stable. However, this matter does not hold on It can be checked that a set in ðX, Ω, PÞ is soft open iff it is the tt-soft semiregular spaces because we replace a partial soft semiopen. Since there exists an unstable soft semiopen nonbelong relation by a total nonbelong relation. There- set in ðX, Ω, PÞ, then ðX, Ω, PÞ is not soft semiregular. There- fore a tt-soft semiregular space needs not be stable. Ω fore, it is not soft semi-T3. However, ðX, , PÞ is a tt-soft semiregular space. Also, it is a tt-soft semi-T -space. Hence, Proposition 37. 1 it is a tt-soft semi-T3-space. Before we show the relationship between tt-soft semi- (i) A tt-soft semi-T -space is always soft semi-T for i = i i Ti-spaces, we need to prove the next useful lemma. 0, 1, 4 Lemma 3. Ω ff (ii) A soft semiregular space is always tt-soft semiregular ðX, , PÞ is a tt-soft semi-T1-space i xP is soft semiclosed for every x ∈ X. (iii) A soft semi-T3-space is always tt-soft semi-T3. ∈ Ω Proof. Necessity: for any distinct point yi X than x, contains a soft semiopen set and G totally contains y so that ⋐ ∈ S iP S i Proof. Since x/GP implies x/GP, then the proofs of (i) and x/⋐G .So ∈ G ðpÞ = X \ fxg and x/⋐ ∈ G ðpÞ for each (ii) follow. iP i I i i I i ~ g ffi p ∈ P. Thus, ∪ ∈ G = X \ fxg is soft semiopen. Hence, To prove (iii), it su ces to prove that a soft semi-Ti-space i I iP Ω is tt-soft semi-Ti when ðX, , PÞ is soft semiregular. Suppose xP is soft semiclosed. 6 Journal of Applied Mathematics

ffi ≠ Su ciency: let x y. By hypothesis, xP and yP are soft In what follows, we found some properties of tt-soft c c semiclosed sets. Then xP and yP are soft semiopen sets so that semi-Ti and tt-soft semiregular. ∈ c ∈ c ⋐ c ⋐ c x ðyPÞ and y ðxPÞ . Obviously, y/ðyPÞ and x/ðxPÞ . Ω Lemma 39. Ω ∈ Hence, ðX, , PÞ is tt-soft semi-T1. Let UP be a subset of ðX, , PÞ and x X. Then ⋐  s ff x/UP i there is a soft semiopen set VP totally containing x Proposition 38. ~ Every tt-soft semi-Ti-space is tt-soft semi- ∩ Φ~ so that UP VP = . Ti−1 for i = 1, 2, 3, 4. ⋐  s ∈  s c ∩~ Φ~ Proof. We suffice by proving the cases of i =3,4. Proof. Let x/UP . Then, x ðUP Þ = VP. So, UP VP = . ≠ Ω Conversely, if there exists a soft semiopen set V totally con- For i =3, let x y in a tt-soft semi-T3-space ðX, , PÞ. P ⋐ Ω taining x so that U ∩~ V = Φ~, then U ⊆ Vc . Therefore, Then, xP is soft semiclosed. Since y/xP and ðX, , PÞ is P P P P Ω  s ⊆ c ⋐ c ⋐  s tt-soft semiregular, then contains two disjoint soft UP VP. Since x/VP, then x/UP . ⊆~ ∈ semiopen sets GP and FP satisfying xP GP and y FP. Therefore, ðX, Ω, PÞ is tt-soft semi-T . Proposition 40. Ω 2 If ðX, , PÞ is a tt-soft semi-T0-space, then ⋐ s s For i =4, let HP be a soft semiclosed set so that x/HP. x ≠ y for every x ≠ y ∈ X. Ω P P Since ðX, , PÞ is tt-soft semi-T1,thenxP is soft semiclosed. ∩~ Φ~ Ω Since xP HP = and ðX, , PÞ is soft seminormal, then ≠ Proof. Let x y in a tt-soft semi-T0-space. Then, there is a there are disjoint soft semiopen sets GP and FP so that soft semiopen set U so that x ∈ U and y/⋐U or y ∈ U ⊆~ ⊆ ~ Ω P P P P HP GP and xP FP. Hence, ðX, , PÞ is tt-soft semi-T3. ⋐ ∈ ⋐ ∩~ Φ~ and x/UP. Say, x UP and y/UP. Now, yP UP = . So, by the above lemma, x/⋐y s. But x ∈ x s. Hence, we obtain the We explain that the converse of proposition above is not P P desired result. always true by presenting the next example. ~ ~ Corollary 41. Ω Example 1. Let P = fp , p g. A family Ω = fΦ, X, fðp , fxgÞ, If ðX, , PÞ is a tt-soft semi-T0-space, then 1 2 1  s y s ∅ Sx ≠ S for all x ≠ y and p, p′ ∈ P. ðp2, Þg is a soft topology on X = fv, wg. The following soft sets p p′ with Ω are all soft semiopen subsets of ðX, Ω, PÞ. Theorem 42. Let P be a finite set and ðX, Ω, PÞ be soft hyper- G = fgðÞp , fgv , ðÞp , fgv Ω 1P 1 2 connected. Then ðX, , PÞ is a tt-soft semi-T1-space if and ∩~ : ∈ ∈ Ωs ∈ only if xP = fUP x UP g for each x X. G = fgðÞp , fgv , ðÞp , fgw 2P 1 2 Proof. To prove the “if” part, let y ∈ X. Then for each x ∈ G = fgðÞp , fgv , ðÞp , X ð3Þ 3P 1 2 ∈ X \ fyg, we have a soft semiopen set UP so that x UP and y/⋐U . Therefore y/⋐∩~fU : x ⊆~ U ∈ Ωsg. Since y is G = fgðÞp , X , ðÞp , fgw P P P P 4P 1 2 chosen arbitrary, then the desired result is proved. To prove the “only if” part, let the given conditions be G = fgðÞp , X , ðÞp , fgw : 5P 1 2 fi ≠ ∣ ∣ ⋐ satis ed and let x y. Let E = m. Since y/xP, then for each ⋯ ∈ j =1,2, , m there is a soft semiopen set Ui so that y/UiðejÞ fð f gÞ ð f gÞg P Since the soft sets p1, v , p2, v are a soft semio- and x ∈ U . Since ðX, Ω, PÞ is soft hyperconnected, then it pen set in ðX, Ω, PÞ,thenitistt-soft semi-T . However, iP 0 ∩~ i=m Ω follows from Theorem 17 that Ui is a soft semiopen fðp1, fwgÞ, ðp2, fwgÞg is not a soft semiopen set in ðX, , i=1 P PÞ.Hence,itisnottt-soft semi-T . set so that y/⋐∩~ i=mU and x ∈∩~ i=mU . Similarly, we can 1 i=1 iP i=1 iP ∈ ⋐ get a soft semiopen set VP so that y VP and x/VP. Thus, Ω Φ~ ⊆~ ℕ : c Ω Example 2. Let = f ,GP GP is f initeg be a soft topol- ðX, , PÞ is a tt-soft semi-T1-space. ogy on the set of natural numbers ℕ, where P is a set of parameters. By a simple check, one notes that a class of soft Theorem 43. If ðX, Ω, PÞ is an extended tt-soft semi-T - ℕ Ω 1 semiopen and a class of soft open sets in ð , , PÞ coincide. x x ∈ ~ g g space, then Sp is soft semiclosed for all Sp X. For each v ≠ w ∈ ℕ, we have ℕ \ fwg and ℕ \ fvg are soft g g semiopen sets so that v ∈ ℕ \ fwg and w/⋐ℕ \ fwg; and w g g g Proof. By Lemma 3, we have X \ fxg is a soft semiopen ∈ ℕ \ fvg and v/⋐ℕ \ fvg. Therefore, ðℕ, Ω, PÞ is tt-soft Ω set. Since ðX, , PÞ is extended, then a soft set HP, where semi-T1. In contrast, all nonnull proper semiopen sets have ∅ ′ ′ ≠ ℕ Ω HðpÞ = and Hðp Þ = X for each p p, is a soft semio- nonnull soft intersection. Hence, ð , , PÞ is not tt-soft ~∪~ pen set. Therefore, X \ fxg HP is soft semiopen. Thus, semi-T2. ~∪~ c x ðX \ fxg HPÞ = Sp is soft semiclosed. Example 3. As we know, the coincidence of soft topological and Corollary 44. Ω classical topological spaces occurs if E is a singleton. Then, it If ðX, , PÞ is an extended tt-soft semi-T1- ffi su ces to consider examples that satisfy a semi-T2-space but space, then the intersection of all soft semiopen sets containing ⊆~ ~ not semi-T3 and satisfy a semi-T3-space but not semi-T4. UP is exactly UP for each UP X. Journal of Applied Mathematics 7

Proof. Let U be a soft subset of X~. Since Sx is a soft semi- Proposition 49. Let ðX, Ω, PÞ be finite soft hyperconnected. If P p Ω for every Sx ∈ Uc , then X~ \ Sx is a soft semiopen ðX, , PÞ is a tt-soft semi-T2-space, then it i tt-soft p P p semiregular. ∩~ ~ x : x ∈ c set containing UP. Therefore, UP = fXSp Sp UPg,as required. ∈ ⋐ Proof. Let HP be a soft semiclosed set and x X so that x/HP. ≠ ⋐ Then x y for each y HP. By hypothesis, there are two dis- Theorem 45. Let ðX, Ω, PÞ be finite soft hyperconnected. joint soft semiopen sets U and V so that x ∈ U and y ∈ Ω iP iP iP Then, ðX, , PÞ is tt-soft semi-T2 if and only if it is tt-soft V . Since fy : y ∈ Xg is a finite set, then there is a finite num- semi-T . iP 1 ber of soft semiopen sets V so that H ⊆~ ∪~ i=mV . Since iP P i=1 iP ðX, Ω, PÞ is soft hyperconnected, then it follows from Proof. Necessity: it is obtained from Proposition 38. ∩~ i=m ffi ≠ Theorem 17 that i=1 Ui is a soft semiopen set containing Su ciency: for each x y, we have xP and yP are soft P fi ∪~ x. Since ½∪~ i=mV Š ∩~ ½ ∩~ i=mU Š = Φ~, then ðX, Ω, PÞ is tt-soft semiclosed sets. Since X is nite, then y∈X\fxgyP and i=1 iP i=1 iP ∪~ Ω semiregular. x∈X\fygxP are soft semiclosed sets. Since ðX, , PÞ is soft ∪~ c ∪~ c hyperconnected, then ð y∈X\fxgyPÞ = xP and ð x∈X\fygxPÞ Corollary 50. Ω = yP are soft semiopen sets. The disjointness of xP and yP The following concepts are identical if ðX, , PÞ Ω fi ends the proof that ðX, , PÞ is tt-soft semi-T2. is nite soft hyperconnected.

(i) a tt-soft semi-T3-space Remark 46. In Example2, note that xP is not a soft semiopen ∈ ℕ fi set for each x . This clari es that a soft set xP in a tt-soft (ii) a tt-soft semi-T2-space semi-T -space need not be soft semiopen if the universal set 1 (iii) a tt-soft semi-T -space. is infinite. 1

Theorem 47. Ω ff ðX, , PÞ is tt-soft semiregular i for every soft Proof. We obtain the directions ðiÞ ⟶ ðiiÞ and ðiiÞ ⟶ ðiiiÞ Ω semiopen subset FP of ðX, , PÞ totally containing x, there is from Proposition 38. ∈ ⊆~  s ⊆~ a soft semiopen set VP so that x VP VP FP. We obtain the direction ðiiiÞ ⟶ ðiiÞ from Theorem 45. We obtain the direction ðiiÞ ⟶ ðiÞ from Proposition 49.

Proof. Let FP be a soft semiopen set totally containing x. Then Fc is semisoft closed and x ∩~ Fc = Φ~.SoΩ contains two Theorem 51. P P P The property of being a tt-soft semi-Ti-space c ⊆~ disjoint soft semiopen sets UP and VP satisfying FP UE ði = 0, 1, 2, 3Þ is a soft open hereditary. ∈ ⊆~ c ⊆~  s ⊆~ c ⊆~ and x VP. Thus, VP UP FP. Hence, VP UP FP. Con- versely, let Fc be a soft semiclosed set. Then, for each x/⋐Fc , P P Proof. We suffice by proving the cases of i =3. we have ∈ . By hypothesis, there is a soft semiopen set x FP Let ðY, Ω , PÞ be a soft open subspace of a tt-soft semi-  s ⊆~ c Y VP totally containing x so that VP FP. Therefore, FP Ω Ω c c T3-space ðX, , PÞ. To prove that ðY, Y , PÞ is tt-soft semi- ⊆~ ðV sÞ and V ∩~ ðV sÞ = Φ~. Thus, ðX, Ω, PÞ is tt-soft ≠ ∈ Ω P P P T1, let x y Y. Since ðX, , PÞ is a tt-soft semi-T1-space, semiregular, as required. then there exist two soft semiopen sets GP and FP so that ∈ ⋐ ∈ ⋐ ∈ x GP and y/GP; and y FP and x/FP. Therefore, x Theorem 48. ~ ∩~ ∈ ~ ∩~ ⋐ ⋐ The three concepts below are identical if ðX, UP = Y GP and y VP = Y FP so that y/UP and x/ Ω , PÞ is a tt-soft semiregular space. VP. It follows from Proposition 18 that UP and VP are Ω Ω soft semiopen subsets of ðY, Y , PÞ, so that ðY, Y , PÞ is (i) a tt-soft semi-T2-space tt-soft semi-T1. Ω ∈ To prove that ðY, Y , PÞ is tt-soft semiregular, let y Y (ii) a tt-soft semi-T1-space Ω ⋐ and FP be a soft semiclosed subset of ðY, Y , PÞ so that y/ ∪~ ec Ω (iii) a tt-soft semi-T0-space. FP. Then, FP Y is a soft semiclosed subset of ðX, , PÞ so ⋐ ∪~ ec that y/FP Y . Therefore, there exist disjoint soft semiopen Ω ∪~ ~ c⊆~ ∈ subsets UP and VP of ðX, , PÞ so that FP Y UP and y Proof. The directions ðiÞ ⟶ ðiiÞ and ðiiÞ ⟶ ðiiiÞ are ∩~ ~ ∩~ ~ VP. Now, UP Y and VP Y are disjoint soft semiopen sub- obvious. Ω ⊆~ ∩~ ~ ∈ ∩~ ~ ⟶ ≠ sets of ðY, Y , PÞ so that FP UP Y and y VP Y. Thus, To prove ðiiiÞ ðiÞ, let x y in a tt-soft semi-T0-space Ω Ω ðY, Y , PÞ is tt-soft semiregular. ðX, , PÞ. Then, there exists a soft semiopen set GP so that Ω ∈ ⋐ ∈ ⋐ ∈ ⋐ Hence, ðY, Y , PÞ is tt-soft semi-T3, as required. x GP and y/GP,ory GP and x/GP. Say, x GP and y/ ⋐ c ∈ c Ω GP. Obviously, x/GP and y GP. Since ðX, , PÞ is tt-soft semiregular, then there exist two disjoint soft semiopen sets Theorem 52. Let ðX, Ω, PÞ be extended and i = 0, 1, 2, 3, 4. ∈ ∈ c ⊆~ Ω Then, ð , Ω, Þ is -soft semi- iff ð , Ω Þ is semi- for UP and VP so that x UP and y GP VP. Hence, ðX, , X P tt Ti X p Ti ∈ PÞ is tt-soft semi-T2. each p P. 8 Journal of Applied Mathematics

≠ ≠ Proof. We prove the theorem in the case of i =4, and one can y1 y2. Without loss of generality, let x1 x2. Then there similarly prove the other cases. exist two disjoint soft semiopen subsets G and H of E1 E1 Necessity: let x ≠ y in X. Then, there exist two soft semio- ðX , Ω , E Þ so that x ∈ G and x ⋐/G ; and x ∈ H and ∈ ⋐ ∈ 1 1 1 1 E1 2 E1 2 E1 pen sets UP and VP so that x UP and y/UP; and y VP and ⋐ f f ⋐ ∈ ∈ ∈ x1 /HE . Obviously, GE × X2 and HE × X2 are two disjoint x/VP. Obviously, x UðpÞ and y/UðpÞ; and y VðpÞ and x 1 1 1 ∈ Ω soft semiopen subsets X × X so that ðx , y Þ ∈ G × Xf /VðpÞ. Since ðX, , PÞ is extended, then it follows from The- 1 2 1 1 E1 2 Ω f f orem 26 that UðpÞ and VðpÞ are semiopen subsets of ðX, pÞ and ðx , y Þ⋐/G × X ; and ðx , y Þ ∈ H × X and ðx , y Þ⋐/ ∈ Ω 2 2 E1 2 2 2 E1 2 1 1 for each p P. Thus, ðX, pÞ is a semi-T1-space. To prove H × Xf. Hence, X × X is a tt-soft semi-T -space. Ω E1 2 1 2 2 that ðX, pÞ is seminormal, let Fp and Hp be two disjoint Ω semiclosed subsets of ðX, pÞ. Let FP and HP be two soft sets For the sake of brevity, we omit the proof of the following ′ ′ ∅ given by FðpÞ = Fp, HðpÞ = Hp, and Fðp Þ = Hðp Þ = for result. each p′ ≠ p. It follows from Theorem 26 that F and H are P P Theorem 54. two disjoint soft semiclosed subsets of ðX, Ω, PÞ.By The property of being a tt-soft semi-Ti-space is an additive property for i = 0, 1, 2, 3, 4. hypothesis, there exist two disjoint soft semiopen sets GP ⊆~ ⊆~ and WP so that FP GP and HP WP. This implies that ⊆ ⊆ Ω We complete this section by discussing some interrela- FðpÞ = Fp GðpÞ and HðpÞ = Hp WðpÞ. Since ðX, , PÞ is extended, then it follows from Theorem 26 that GðpÞ tions between tt-soft semi-Ti-spaces ði =2,3,4Þ and soft Ω Ω semicompact spaces. and WðpÞ are semiopen subsets of ðX, pÞ. Thus, ðX, pÞ is a seminormal space. Hence, it is a semi-T -space. 4 Proposition 55. A stable soft semicompact set in a tt-soft Sufficiency: let x ≠ y in X. Then there exists two semiopen Ω ∈ ∈ semi-T2-space is soft semiclosed. subsets Up and Vp of ðX, pÞ so that x Up and y/Up; and ∈ ∈ y Vp and x/Vp. Let UP and VP be two soft sets given by Proof. It follows from Proposition 22 and Remark 35. ∈ Ω UðpÞ = Up and VðpÞ = Vp for each p P. Since ðX, , PÞ is Theorem 56. extended, then it follows from Theorem 26 that UP and VP Let HP be a soft semicompact subset of a soft ð Ω Þ ∈ ⋐ are soft semiopen subsets of X, , P so that x UP and y hyperconnected tt-soft semi-T2-space. If x/HP, then there ⋐ ∈ ⋐ ð Ω Þ ∈ /UP; and y VP and x/VP. Thus, X, , P is a tt-soft are disjoint soft semiopen sets UP and VP so that x UP and ð Ω Þ ⊆ semi-T1-space. To prove that X, , P is soft seminormal, HP VP. let FP and HP be two disjoint soft semiclosed subsets of ðX, Ω, PÞ. Since ðX, Ω, PÞ is extended, then it follows from ⋐ ≠ ⋐ Ω Proof. Let x/HP. Then x y for each y HP. Since ðX, , PÞ Theorem 26 that FðpÞ and HðpÞ are two disjoint semiclosed is a tt-soft semi-T -space, then there exist disjoint soft semi- Ω 2 subsets of ðX, pÞ. By hypothesis, there exist two disjoint open sets U and V so that x ∈ U and y ∈ V . Therefore, Ω ⊆ iP iP iP iP semiopen subsets Gp and Wp of ðX, pÞ so that FðpÞ Gp fV g forms a soft semiopen cover of H . Since H is soft ⊆ iP P P and HðpÞ Wp. Let GP and WP be two soft sets given by G semicompact, then H ⊆∪~i=nV . By the soft hyperconnect- ∈ Ω P i=1 TiP ðpÞ = Gp and WðpÞ = Wp for each p P. Since ðX, , PÞ is e i=n edness of ðX, Ω, PÞ, we obtain U = U is a soft semio- i=1 iP P extended, then it follows from Theorem 26 that GP and WP are two disjoint soft semiopen subsets of ðX, Ω, PÞ so that pen set. Hence, we obtain the desired result. F ⊆~ G and H ⊆~ W . Thus, ðX, Ω, PÞ is soft seminormal. P P P P Theorem 57. Every soft hyperconnected, soft semicompact, Hence, it is a tt-soft semi-T4-space. and tt-soft semi-T2-space is tt-soft semiregular. In the following examples, we show that a condition of an extended soft topology given in the above theorem is not Proof. Let HP be a soft semiclosed subset of soft semicompact Ω ⋐ superfluous. and tt-soft semi-T2-space ðX, , PÞ so that x/HP. Then, HP is soft semicompact. By Theorem 56, there exist disjoint soft ∈ ⊆ Example 1. Let ðX, Ω, PÞ be the same as in Example1. We semiopen sets UP and VP so that x UP and HP VP. Thus, Ω ðX, Ω, PÞ is tt-soft semiregular. showed that ðX, , PÞ is not tt-soft semi-T0. On the other hand, Ω and Ω are the discrete topology on X. Hence, p1 p2 Corollary 58. Every soft hyperconnected, soft semicompact, the two parametric topological spaces ðX, Ω Þ and ðX, Ω Þ p1 p2 and tt-soft semi-T2-space is tt-soft semi-T3. are semi-T4. Lemma 59. Theorem 53. Let FP be a soft semiopen subset of a soft semire- The property of being a tt-soft semi-Ti-space gular space. Then, for each Sx ∈ F , there exists a soft semiopen ði = 0, 1, 2Þ is preserved under a finite product soft spaces. p P x ∈  s ⊆~ set GP so that Sp GP FP. Proof. We prove the theorem in case of i =2. The other cases x ∈ follow similar lines. Proof. Let FP be a soft semiopen set so that Sp FP. Then, Ω Ω ∈ c Ω Let ðX1, 1, E1Þ and ðX2, 2, E2Þ be two tt-soft semi-T2- x/FP. Since ðX, , PÞ is soft semiregular, then there exist ≠ ≠ spaces and let ðx1, y1Þ ðx2, y2Þ in X1 × X2. Then, x1 x2 or two disjoint soft semiopen sets GP and WP totally containing Journal of Applied Mathematics 9

c ∈ ⊆~ c ⊆~ ∗ x and FP, respectively. Thus, x GP WP FP. Hence, semi -continuous map. Then there exists a unique soft point x ∈ ⊆~  s ⊆~ c ⊆~ x ∈ ~ Sp GP GP WP FP. Sp X of gφ.

Theorem 60. B ~ B B n ~ Let HP be a soft semicompact subset of a soft Proof. Let f 1 = gφðXÞ and n = gφð n−1Þ = gφðXÞ for semiregular space and FP be a soft semiopen set containin each n ∈ ℕg be a family of soft subsets of ðX, Ω, PÞ. It is clear ~ ∗ HP. Then, there exists a soft semiopen set GP so that HP that B ⊆ B for each n ∈ ℕ. Since gφ is soft semi -con- ~ ~  s ~ n+1 n ⊆ G ⊆ G ⊆ F . B ∈ ℕ P P P tinuous, then n is a soft semicompact set for each n Ω ′ B and since ðX, , PÞ is soft semi-T 2, then n is also a soft Proof. Let the given conditions be satisfied. Then, for each semiclosed set for each n ∈ ℕ. It follows from Theorem Sx ∈ H , we have Sx ∈ F . Therefore, there is a soft semiopen ∩~ B p P p P 62 that ðH, PÞ = n∈ℕ n is a nonnull soft set. Note that set W so that Sx ∈ W ⊆~ W s ⊆~ F . Now, fW : Sx ∈ ∩~ n ~ ⊆~ ∩~ n+1 ~ ⊆~ ∩~ n ~ xeP p xeP xeP P xeP p gφðH, PÞ = gφð n∈ℕgφðX ÞÞ n∈ℕgφ ðX Þ n∈ℕgφðXÞ ~ FPg is a soft semiopen cover of HP. Since HP is soft semicom- = ðH, PÞ. To show that ðH,PÞ ⊆ gφðH, PÞ, suppose that ~ ~ i=n ~ i=n pact, then H ⊆ ∪ W . Putting G = ∪ W . Thus, x ∈ x∈ C −1 P i=1 xeP P i=1 xeP there is a Sp ðH, PÞ so that Sp /gφðH, PÞ. Let n = gφ ~ ~  s ~ H ⊆ G ⊆ G ⊆ F . x ∩~ B C ≠ Φ~ C ⊆~ C ∈ ℕ P P P P ðSpÞ n.Obviously, n and n n−1 for each n . −1 x ′ Now, gφ ðS Þ is a soft compact subset of a soft semi-T -space, Corollary 61. If ðX, Ω, PÞ is soft semicompact and soft semi- p 2 −1 x C T , then it is tt-soft semi-T . then gφ ðSpÞ is soft closed; therefore, n is a soft semiclosed set 3 4 ∈ ℕ y for each n .ByTheorem62,thereexistsasoftpointPm so y −1 x ~ x y that P ∈ gφ ðS Þ ∩ B . Therefore, S = gφðP Þ ∈ gφðH, PÞ. Proof. Suppose that F1 and F2 are two disjoint soft semi- m p n p m ~P P closed sets. Then, F ⊆ Fc . Since ðX, Ω, PÞ is soft semicom- This is a contradiction. Thus, gφðH, PÞ = ðH, PÞ. Hence, the 2P 1P pact, then F is soft semicompact and since ðX, Ω, PÞ is soft proof is complete. 2P semiregular, then there is a soft semiopen set GP so that ~ ~ s ~ ~ ~ s c Definition 64. F ⊆ G ⊆ G ⊆ Fc . Obviously, F ⊆ G , F ⊆ ðG Þ and 2P P P 1P 2P P 1P P ∩~  s c Φ~ Ω GP ðGP Þ = . Thus, ðX, , PÞ is soft seminormal. Since (i) ðX, Ω, PÞ is said to have a semifixed soft point Ω ∗ ðX, , PÞ is soft semi-T3, then it is tt-soft semi-T1. Hence, property if every soft semi -continuous map gφ : it is tt-soft semi-T4. ðX, Ω, PÞ ⟶ ðX, Ω, PÞ has a fixed soft point ∗ 4. Semifixed Soft Points of Soft Mappings (ii) A property is said to be a semi -soft if the property is preserved by soft semi∗- In this section, we introduce a semifixed soft point property homeomorphism maps. and investigate some main features, in particular, those related to parametric topological spaces. Proposition 65. The property of being a semifixed soft point is ∗ Theorem 62. B : ∈ ℕ a semi -soft topological property. Let f n n g be a collection of soft subsets of a soft semicompact space ðX, Ω, PÞ satisfying the following: Proof. Let ðX, Ω, PÞ and ðY, θ, PÞ be a soft semi∗-homeo- B ≠ Φ~ ∈ ℕ morphic. Then, there is a bijective soft map f φ : ðX, Ω, PÞ (i) n for each n −1 ∗ ⟶ ðY, θ, PÞ so that f φ and f φ are soft semi -continuous. (ii) B is a soft semiclosed set for each n ∈ ℕ n Since ðX, Ω, PÞ has a semifixed soft point property, then B ⊆~ B ∈ ℕ ∗ : Ω ⟶ Ω (iii) n+1 n for each n . every soft semi -continuous map gφ ðX, , PÞ ðX, , PÞ has a semifixed soft point. Now, let hφ : ðY, θ, PÞ ⟶ ~ ~ ðY, θ, PÞ be a soft semi∗-continuous. Obviously, hφ ∘ f φ : Then, ∩ ∈ℕB ≠ Φ. n n ðX, Ω, PÞ ⟶ ðY, θ, PÞ is a soft semi∗-continuous. Also, −1 ∘ ∘ : Ω ⟶ Ω ∗ ∩~ B Φ~ ∪~ Bc ~ f φ hφ f φ ðX, , PÞ ðX, , PÞ is a soft semi -con- Proof. Suppose that n∈ℕ n = . Then, n∈ℕ n = X.It Ω fi follows from (ii) that fBc : n ∈ ℕg is a soft semiopen cover tinuous. Since ðX, , PÞ has a semi xed soft point property, n −1 x x x ∈ ~ ~ then f φ ðhφðf φðSpÞÞÞ = Sp for some Sp X. consequently, f φ of X. By hypothesis of soft semicompactness, there exist i1, − ~ c ~ c ~ 1 x x x i , ⋯, i ∈ ℕ, i < i < ⋯ < i so that X = B ∪ B ∪ ⋯ ðf φ ðhφðf φðSpÞÞÞÞ = f φðSpÞ. This implies that hφðf φðSpÞÞ = 2 k 1 2 k i1 i2 ~ ~ ~ ~ x x fi ∪ Bc B ⊆ ~ Bc ∪ Bc ∪ ⋯ f φðSpÞ. Thus, f φðSpÞ is a semi xed soft point of hφ. Hence, . It follows from (iii) that i X = ik k i1 i2 θ fi ∪~ Bc = ½B ∩~ B ∩~ ⋯∩~ B Šc = Bc . This yields a contra- ðY, , PÞ has a semi xed soft point property, as required. ik i1 i2 ik ik ∩~ B ≠ Φ~ diction. Thus, we obtain the proof that n∈ℕ n . Before we investigate a relationship between soft topolog- ical space and their parametric topological spaces in terms of Proposition 63. Let ðX, Ω, PÞ be a soft semicompact and soft possessing a fixed (soft) point, we need to prove the following ′ : Ω ⟶ Ω semi-T 2-space and gφ ðX, , PÞ ðX, , PÞ be a soft result. 10 Journal of Applied Mathematics

Theorem 66. If Ω is extended, then gφ : ðX, Ω, PÞ ⟶ ðY, θ eral, we have studied their main properties and showed the ∗ , PÞ is soft semi -continuous iff g : ðX, Ω Þ ⟶ ðY, θϕ Þ is interrelations between them with help of interesting exam- p ðpÞ fi fi semi∗-continuous. ples. Second, we have de ned semi xed soft point and inves- tigated its basic properties. θ As future works, we shall study them on the contents of Proof. Necessity: let U be a semiopen subset of ðY, ϕðpÞÞ. θ ϕ supra soft topological spaces, minimal soft topological Then contains a soft semiopen set GP satisfying Gð ðpÞÞ ∗ −1 spaces, and soft weak structures. Finally, we hope that the = U.Sincegφ is a soft semi -continuous map, then gϕ ðGPÞ concepts initiated herein will find their applications in many is a soft semiopen set. Definition 8 implies that a soft subset fields soon. −1 −1 −1 gϕ ðG Þ = ðgϕ ðGÞÞ of ðX, Ω, PÞ is given by gϕ ðGÞðpÞ = P P − g 1ðGðϕðpÞÞÞ for each p ∈ P.Byhypothesis,Ω is extended; Data Availability we obtain from Theorem 26 that a subset g−1ðGðϕðpÞÞÞ = −1 Ω ∗ g ðUÞ of ðX, pÞ is semiopen. Hence, a map g is semi - No data were used to support this study. continuous. ffi θ Su ciency: let GP be a soft semiopen subset of ðY, , PÞ. Conflicts of Interest fi −1 According to De nition 8, a soft subset gϕ ðGPÞ = −1 −1 −1 ðgϕ ðGÞÞ of ðX, Ω, PÞ is given by gϕ ðGÞðpÞ = g ðGðϕðpÞÞ The author declares that he has no competing interests. P Þ for each p ∈ P. Since a map g is semi∗-continuous, then a −1 ϕ Ω subset g ðGð ðpÞÞÞ of ðX, pÞ is semiopen. By hypothesis, Acknowledgments −1 Ω is extended; we obtain from Theorem 26 that gϕ ðG Þ is P We are highly grateful to the anonymous referees for their ð Ω Þ a soft semiopen subset of X, , P . Hence, a soft map gφ is helpful comments and suggestions for improving the paper. soft semi∗-continuous.

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