@ Appl. Gen. Topol. 22, no. 1 (2021), 17-30doi:10.4995/agt.2021.13084 © AGT, UPV, 2021 On soft quasi-pseudometric spaces Hope Sabao a and Olivier Olela Otafudu b a School of Mathematics, University of the Witwatersrand Johannesburg 2050 , South Africa. (hope.sabao@wits.ac.za) b Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville 7535, South Africa. (olmaolela@gmail.com) Communicated by S. Romaguera Abstract In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compati- ble quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the con- cept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set. 2010 MSC: 03E72; 08A72; 47H10; 54E35; 54E15. Keywords: soft-metric; soft-quasi-pseudometric; soft Isbell convexity. 1. Introduction Soft set theory has several applications in solving practical problems in eco- nomics, engineering, social sciences and medical science e.t.c. The study of soft sets was first initiated by Molodtsov [8] in 1999. Since then, many other scholars have taken interest in soft set theory (See [1], [2], [4], [15] ). The study of soft metric spaces was initiated by Das and Samanta in [16]. Using the concept of a soft point in a soft set, they introduced a soft metric and some of their basic properties. Thereafter, they investigated some topological structures such as soft open sets, soft closed sets and soft closures of soft sets e.t.c. Furthermore, they investigated the notion of completeness of soft metric Received 31 January 2020 – Accepted 19 October 2020 http://dx.doi.org/10.4995/agt.2021.13084 H. Sabao and O. Olela Otafudu spaces and the Cantor’s Intersection Theorem. Recently, Abbas et al. [5] studied the concept of fixed point theory of soft metric spaces. They showed that a soft metric induces a compatible metric on the collection of all soft points of the absolute soft set, whenever the set of parameters is finite. Thereafter, they used this concept to show that several fixed point theorems for metric spaces can be directly deduced from compara- ble existing results. Until recently, most studies in topology has been based on spaces arising from a collection of metrics, which like the Euclidean distance, are symmetric. This was very natural since many problems it was used for were based on the Eu- clidean topology on the real numbers, which arises from the usual distance on reals numbers. But most topologies are not distance based, they are based on things like “effort” which have many properties of metrics but lack symmetry. A quasi-metric space is an example of a space which lack symmetry. It is also well known that quasi-metric spaces constitute an efficient tool to discuss and solve several problems in topological algebra, approximation theory, theoretical computer science, etc. (see [10]). On the other hand, T -theory is a theory that involves trees, injective envelopes of metric spaces (hyperconvex hull), and all of the areas that are connected with these topics. These topics have been used in the development of mathematical tools for reconstructing phylogenetic trees (see [6]). These are our motivations for generalising soft metric spaces to the asymmetric setting and introducing the concept of hyperconvexity in our new space. In this article, we introduce soft quasi-pseudometric spaces, a concept that generalise soft metric spaces to the asymmetric setting. We then show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of hyperconvexity in soft quasi- pseudometric spaces, which we call soft Isbell convexity, and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set. 2. Preliminaries The letters U, E and P(U) will denote the universal set, the set of parameters and the power set of U respectively. According to [8], if F is a set valued mapping on A ⊂ E taking values in P(U), then the pair (F, A) is called a soft set over U. We will denote the collection of soft sets over a common universe U by S(U). A soft set (F, A) over U is said to be a soft point if there is exactly one λ ∈ A such that F(λ) = {x} and F(e) = ∅ for all e ∈ A \ {λ}. We shall denote such © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 18 On soft quasi-pseudometric spaces a point by (F xλ , A) or simply F x λ . A soft point F x λ is said to belong to a soft set (F, A), denoted by F xλ ∼ ∈ (F, A), if F xλ (λ) = {x} ⊂ F(λ). The collection of soft points of (F, A) is denoted by SP(F, A). A soft set (F, E) is said to be a null soft set, denoted by Φ if for all e ∈ A, F(e) = ∅. A soft set which is not null is said to be a non-null soft set. f is a soft mapping from the soft set (F, A) to a soft set (G, B), denoted by f : (F, A) ∼ −→ (G, B), if for each soft point F xλ ∼ ∈ (F, A) there exists only one soft point Gyµ ∼ ∈ (G, B) such that f(F xλ ) = G y µ. Now let R be the set of real numbers. We denote the collection of all nonempty bounded subsets of R by B(R). A soft real set, denoted by (f̂, A) or simply f̂ is a mapping f̂ : A → B(R). If f̂ is a single valued mapping on A ⊂ E taking values in R, then the pair (f̂, A) or simply f̂ is called a soft element of R or a soft real number. If f̂ is a single valued mapping on A ⊂ E taking values in the set R+ of nonnegative real numbers, then the pair (f̂, A), or simply f̂, is called a nonnegative soft real number. We shall denote the set of nonnegative soft real numbers by R(A)∗. A constant soft real number c is a soft real number such that for each e ∈ A, we have c(e) = c, where c is some real number. Definition 2.1 ([8]). For two soft real numbers f̂, ĝ we say that (i) f̂ ∼ ≤ ĝ if f̂(e) ≤ ĝ(e) for all e ∈ A (ii) f̂ ∼ ≥ ĝ if f̂(e) ≥ ĝ(e) for all e ∈ A (iii) f̂ ∼ < ĝ if f̂(e) < ĝ(e) for all e ∈ A (iv) f̂ ∼ > ĝ if f̂(e) > ĝ(e) for all e ∈ A Definition 2.2. Let U be a universal set, A be a nonempty subset of pa- rameters and ∼ U be the absolute soft set, i.e F(λ) = U for all λ ∈ A, where (F, A) = ∼ U. A mapping d : SP( ∼ U) × SP( ∼ U) → R(A)∗ is said to be a soft pseu- dometric on ∼ U if for any Uxλ , U y µ, U z λ ∈ SP( ∼ U) (equivalently Uxλ , U y µ, U z λ ∼ ∈ ∼ U), the following hold: (i) d(Uxλ , U x λ ) = 0 (ii) d(Uxλ , U y µ) = d(U y µ, U x λ ) (iii) d(Uxλ , U z λ) ∼ ≤ d(Uxλ , U y µ) + d(U y µ, U z λ) The soft set ∼ U endowed with a soft pseudometric d is called a soft pseudometric space and is denoted by ( ∼ U, d, A), or simply by ( ∼ U, d) if no confusion arises. Definition 2.3. Let U be a universal set, A be a nonempty subset of pa- rameters and ∼ U be the absolute soft set, i.e F(λ) = U for all λ ∈ A, where (F, A) = ∼ U. A mapping d : SP( ∼ U) × SP( ∼ U) → R(A)∗ is said to be a soft metric on ∼ U if for any Uxλ , U y µ, U z λ ∈ SP( ∼ U) (equivalently Uxλ , U y µ, U z λ ∼ ∈ ∼ U), the following hold: © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 19 H. Sabao and O. Olela Otafudu (i) d(Uxλ , U y µ) = 0 iff U x λ = U y µ (ii) d(Uxλ , U y µ) = d(U y µ, U x λ ) (iii) d(Uxλ , U z λ) ∼ ≤ d(Uxλ , U y µ) + d(U y µ, U z λ) The soft set ∼ U endowed with a soft metric d is called a soft metric space and is denoted by ( ∼ U, d, A), or simply by ( ∼ U, d) if no confusion arises. 3. Soft quasi-pseudometric spaces In this section, we introduce the concept of a soft quasi-pseudometric space. We show that the symmetrised soft pseudometric coincides with the soft pseu- dometric in the sense of [16]. Definition 3.1. Let U be a universal set, A be a nonempty subset of param- eters and ∼ U be the absolute soft set. A mapping q : SP( ∼ U) × SP( ∼ U) → R(A)∗ is said to be a soft quasi-pseudometric on ∼ U if for any Uxλ , U y µ, U z λ ∈ SP( ∼ U) (equivalently Uxλ , U y µ, U z λ ∼ ∈ ∼ U), the following hold: (i) q(Uxλ , U x λ ) = 0 (ii) q(Uxλ , U z λ) ∼ ≤ q(Uxλ , U y µ) + q(U y µ, U z λ). We say q is a soft quasi-metric provided that q also satisfies the following condition: q(Uxλ , U y µ) = 0 = q(U y µ, U x λ ) implies U x λ = U y µ. The soft set ∼ U endowed with a soft quasi-pseudometric is called a soft quasi pseudometric space denoted by ( ∼ U, q, A) or simply by ( ∼ U, q) if no confusion arises. Remark 3.2. If q is a soft quasi-pseudometric (soft quasi-metric) on ∼ U, then qt : SP( ∼ U) × SP( ∼ U) → R(A)∗ and qs : SP( ∼ U) × SP( ∼ U) → R(A)∗ defined by qt(Uxλ , U y µ) = q(U y µ, U x λ ) and q s(Uxλ , U y µ) = max{q(U x λ , U y µ), q t(Uxλ , U y µ)} are also a soft quasi-pseudometric (soft quasi-metric) and soft pseudometric (soft metric) on ∼ U respectively. Note that qs is a soft (pseudometric) metric in the sense of [16]. Furthermore, qt is called the conjugate of q. Furthermore, we have q(Uxλ , U y µ) ≤ q s(Uxλ , U y µ) and q t(Uxλ , U y µ) ≤ q s(Uxλ , U y µ). Definition 3.3. Let ( ∼ U, q) be a soft quasi-pseudometric space and r̂ be a non- negative soft real number. For any Uxλ ∼ ∈ ∼ U, we define the open and closed balls with radius r̂ and center Uxλ respectively as follows: Bq(U x λ , r̂) = {U y µ ∼ ∈ ∼ U : q(Uxλ , U y µ) ∼ < r̂} and Cq(U x λ , r̂) = {U y µ ∼ ∈ ∼ U : q(Uxλ , U y µ) ∼ ≤ r̂}. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 20 On soft quasi-pseudometric spaces SS(Bq(U x λ , r̂)) is called the soft open ball with center U x λ and radius r̂ while SS(Cq(U x λ , r̂)) is called the soft closed ball with center U x λ and radius r̂. Example 3.4. Let U ⊆ R be a non-empty set and A ⊂ R be the non-empty set of parameters. Let ∼ U be the absolute soft set, that is, F(λ) = U ∀λ ∈ A, where (F, A) = ∼ U. Let x denote the soft real number such that x(λ) = x for all λ ∈ A. Furthermore, for constant soft real numbers x and y, put x−̇y = max{x−y, 0}. Then q : SP( ∼ U) × SP( ∼ U) −→ R(A)∗, defined by q(Uxλ , U y µ) = x−̇y + λ−̇µ, is a soft quasi-metric. Proof. (i) q(Uxλ , U x λ ) = x−̇x + λ−̇λ = 0 (ii) q(Uxλ , U y µ) = q(U y µ, U x λ ) = 0 =⇒ x−̇y + λ−̇µ = 0 =⇒ x−̇y = 0 and λ−̇µ = 0 =⇒ x = y and λ = µ =⇒ Uxλ = U y µ. (iii) q(Uxλ , U y µ) = x−̇y + λ−̇µ = x−̇y + z−̇z + λ−̇µ + γ−̇γ ∼ ≤ x−̇z + z−̇y + λ−̇γ + γ−̇µ ∼ ≤ q(Uxλ , U z γ ) + q(U z γ , U y µ). Therefore, ( ∼ U, q, A) is a soft quasi-metric space. � Remark 3.5. Notice in the example above that qt : SP( ∼ U)×SP( ∼ U) −→ R(A)∗ defined by qt(Uxλ , U y µ) = q(U y µ, U x λ ) is also a soft quasi-metric on ∼ U. Further- more, qs : SP( ∼ U) × SP( ∼ U) −→ R(A)∗ defined by qs(Uxλ , U y µ) = max{q(U x λ , U y µ), q t(Uxλ , U y µ)} = |x − y| + |λ − µ| is a soft metric on ∼ U in the sense of [16]. Proposition 3.6. Let ( ∼ U, q) be a soft quasi-pseudometric space and Uxλ ∼ ∈ ∼ U. Then we have the following: (i) Bqs(U x λ , r̂) ⊆ Bq(U x λ , r̂) (ii) Cqs(U x λ , r̂) ⊆ Cq(U x λ , r̂) Definition 3.7. A soft subset (Y, A) in a soft quasi-pseudometric space ( ∼ U, q, A) is said to be τ(q)-soft open if for any soft point Uxλ of (Y, A), there exists a positive soft real number r̂ such that Uxλ ∈ Bq(U x λ , r̂) ⊂ SP(Y, A). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 21 H. Sabao and O. Olela Otafudu Remark 3.8. The collection τ(q) of all τ(q)-soft open sets in a soft quasi- pseudometric space ( ∼ U, q) form a τ(q)-soft topology on ∼ U. Similarly, the col- lection τ(qt) of all τ(qt)-soft open sets in a soft quasi-pseudometric space ( ∼ U, q) form a τ(qt)-soft topology on ∼ U. Furthermore, the collection τ(qs) of all τ(qs)- soft open sets in a soft quasi-pseudometric space ( ∼ U, q) form a τ(qs)-soft topol- ogy on ∼ U. The soft topology τ(qs) is finer than the soft topologies τ(q) and τ(qt). Finally the triple (X, τ(q), τ(qt)) is a soft bitopological space. Definition 3.9. Let ( ∼ U, q) be a soft quasi-pseudometric space. A sequence (Uxn λn )n∈N of soft points in ∼ U is said to be τ(q)-convergent in ( ∼ U, q) if there is a soft point Uyµ ∼ ∈ ∼ U such that q(Uxn λn , U y µ) → 0 as n → ∞. Definition 3.10. Let ( ∼ U, q) be a soft quasi-pseudometric space. A sequence {Uxi λi }i∈N of soft points in ∼ U is said to be τ(qt)-convergent in ( ∼ U, q) if there is a soft point Uyµ ∼ ∈ ∼ U such that qt(Uxn λn , Uyµ) = q(U y µ, U xn λn ) → 0 as n → ∞. Proposition 3.11. Let (Uxn λn )n∈N be a sequence in a soft quasi-pseudometric space ( ∼ U, q). Then (i) if (Uxn λn )n∈N is τ(q)-convergent to U x λ and τ(q t)-convergent to Uyµ, then q(Uxλ , U y µ) = 0. (ii) if (Uxn λn )n∈N is τ(q)-convergent to U x λ and q(U y µ, U x λ ) = 0, then (U xn λn )n∈N is τ(q)-convergent to Uyµ. Proof. (i) By letting n → ∞ in the inequality q(Uxλ , U y µ) ∼ ≤ q(Uxλ , U xn λn )+q(Uxn λn , Uzµ), we get q(Uxλ , U y µ) = 0. (ii) Follows from the relation q(Uyµ, U xn λn ) ∼ ≤ q(Uyµ, U x λ ) + q(U x λ , U xn λn ) → 0 as n → ∞. � Definition 3.12. A sequence (Uxn λn )n∈N of soft points in a soft metric space ( ∼ U, d) is said to be Cauchy in ( ∼ U, d) if for each ǫ̂ ∼ ≥ 0, there exists an m ∈ N such that d(Uxi λi , U xj λj ) ∼ < ǫ̂ for all i, j ≥ m. Definition 3.13. A soft metric space ( ∼ U, d) is said to be complete if every Cauchy sequence in ( ∼ U, d) converges to some soft point of ∼ U. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 22 On soft quasi-pseudometric spaces Definition 3.14. A soft quasi-metric space ( ∼ U, q) is said to be bicomplete provided that ( ∼ U, qs) is a complete soft metric space. 4. The compatible quasi-pseudometric In [5], Abbas et al. introduced the concept of a compatible soft metric and used this concept to prove some fixed point theorems. In this section, we introduce the concept of a compatible soft quasi-pseudometric metric whose symmetrised (pseudo) metric coincides with the compatible metric in the sense of [5]. Theorem 4.1. Let ( ∼ U, q, A) be a soft quasi-pseudometric with A a finite set. Define a function Mq : SP( ∼ U) × SP( ∼ U) → R+ as Mq(U x λ , U y µ) = max η∈A q(Uxλ , U y µ)(η) for all Uxλ , U y µ ∈ SP( ∼ U). Then the following holds: (i) Mq is a quasi-pseudometric on SP( ∼ U) (ii) Mq is a quasi-metric on SP( ∼ U) if and only if q is a soft quasi-metric on ∼ U (iii) ( ∼ U, q, A) is bicomplete if and only if (SP( ∼ U), Mq) is bicomplete. Proof. Let Uxλ , U y µ, U z γ ∈ SP( ∼ U). Then we have (i) We first show that Mq satisfies the conditions of a quasi-pseudometric. (i) Mq(U x λ , U x λ ) = 0 by condition (i) of Definition 3.1. (ii) Mq(U x λ , U z γ )+Mq(U z γ , U z µ) by condition (ii) of Definition 3.1. This is because Mq(U x λ , U y µ) = max η∈A q(Uxλ , U y µ) ≤ max η∈A q(Uxλ , U z γ ) + max η∈A q(Uzγ , U y µ) = Mq(U x λ , U z γ ) + Mq(U z γ , U z µ). Therefore, ( ∼ U, Mq) is a quasi-pseudometric space. (ii) If q is a quasi-metric on ∼ U, then Mq(U x λ , U y µ) = Mq(U y µ, U x λ ) = 0 =⇒ U x λ = U y µ. (iii) Suppose ( ∼ U, q, A) is bicomplete. Then ( ∼ U, qs, A) is complete. Then by [5, Theorem 1], (SP( ∼ U), (Mq) s) is complete and so (SP( ∼ U), Mq) is bicomplete. Conversely, suppose (SP( ∼ U), Mq) is bicomplete. Then (SP( ∼ U), (Mq) s) is complete. Thus by [5, Theorem 1], ( ∼ U, qs, A) is complete. Therefore, ( ∼ U, q, A) is bicomplete. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 23 H. Sabao and O. Olela Otafudu Proposition 4.2. Let ( ∼ U, d, A) be a soft quasi-pseudometric space. Then the function (Mq) t : SP( ∼ U) × SP( ∼ U) → R+ defined by (Mq) t(Uxλ , U y µ) = Mq(U y µ, U x λ ) is also a quasi-pseudometric on SP( ∼ U). Moreover, (Mq) t = Mqt. Proof. One can easily check that (Mq) satisfies the axioms of a quasi-pseudometric. We only show that (Mq) t = Mqt. Observe that (Mq) t(Uxλ , U y µ) = Mq(U y µ, U x λ ) = max η∈A q(Uyµ, U x λ )(η) = max η∈A qt(Uxλ , U y µ)(η) = Mqt � Proposition 4.3. Let ( ∼ U, q, A) be a soft quasi-pseudometric space. Then for any soft point Uxλ of ∼ U the following holds: (i) Bq(U x λ , r) = BMq (U x λ , r) and Bqt(U x λ , r) = BMqt (U x λ , r) (ii) Cq(U x λ , r) = CMq (U x λ , r) and Cqt (U x λ , r) = CMqt (U x λ , r) Proof. We show that Bq(U x λ , r) ⊆ BMq (U x λ , r), the rest follows the same ar- guments. Suppose Uyµ ∈ Bq(U x λ , r). Then q(U x λ , U y µ) ∼ < r. This implies that q(Uxλ , U y µ)(η) < r(η) for all η ∈ A. Thus max η∈A q(Uxλ , U y µ)(η) < r. Therefore, Mq(U x λ , U y µ) < r and so Bq(U x λ , r) ⊆ BMq (U x λ , r). Conversely, suppose U y µ ∈ BMq (U x λ , r), then Mq(U x λ , U y µ) < r this implies that max η∈A q(Uxλ , U y µ)(η) < r. Therefore, q(Uxλ , U y µ)(η) < r(η) for all η ∈ A. Hence q(U x λ , U y µ) ∼ < r and so BMq (U x λ , r) ⊆ Bq(U x λ , r) � 5. Soft Isbell convexity In this section, we extend the concept of Isbell convexity, introduced in [13] to soft quasi-pseudometric spaces. Definition 5.1. A soft quasi-pseudometric space ( ∼ U, q) is said to be soft Isbell convex provided that for each family (Uxi λi )i∈I of soft points of ∼ U and fami- lies (ri)i∈I and (si)i∈I of constant non-negative soft real numbers satisfying q(Uxi λi , U xj λj ) ∼ ≤ ri + sj whenever i, j ∈ I, the following holds: ∼⋂ i∈I SS(Cq(U xi λi , ri)) ∼ ∩ SS(Cqt(U xi λi , si)) 6= Φ. or equivalently ⋂ i∈I Cq(U xi λi , ri) ∩ Cqt(U xi λi , si) 6= ∅. Lemma 5.2. Suppose ( ∼ U, q, A), where A is finite, is a soft quasi-pseudometric space. Then ( ∼ U, q, A) is soft Isbell convex if and only if (SP( ∼ U), Mq) is Isbell convex. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 24 On soft quasi-pseudometric spaces Proof. Suppose ( ∼ U, q, A) is soft Isbell convex. Let (Uxi λi )i∈I be a family of soft points of ∼ U and (ri)i∈I and (si)i∈I be families of non-negative real numbers satisfying Mq(U xj λi , U xj λi ) ≤ ri + sj whenever i, j ∈ I. Then max η∈A q(U xj λi , U xj λi )(η) ≤ ri + sj whenever i, j ∈ I. Thus q(U xj λi , U xj λi )(η) ≤ ri + sj for all η ∈ A and i, j ∈ I. Thus q(U xj λi , U xj λi ) ∼ ≤ ri + sj whenever i, j ∈ I. Since ( ∼ U, q, A) is soft Isbell convex, we have ∼⋂ i∈I SS(Cq(U xi λi , ri)) ∼ ∩ SS(Cqt(U xi λi , si)) 6= Φ. or equivalently ⋂ i∈I Cq(U xi λi , ri) ∩ Cqt(U xi λi , si) 6= ∅. Then by Proposition 4.3, we have ⋂ i∈I CMq (U xi λi , ri) ∩ CM qt (Uxi λi , si) = ⋂ i∈I Cq(U xi λi , ri) ∩ Cqt(U xi λi , si) 6= ∅. Therefore, (SP( ∼ U), Mq) is Isbell convex. Conversely, suppose (SP( ∼ U), Mq) is Isbell convex. Let (U xi λi )i∈I be a fam- ily of soft points of ∼ U and (ri)i∈I and (si)i∈I be families of non-negative soft real numbers satisfying q(U xj λi , U xj λi ) ∼ ≤ ri + sj whenever i, j ∈ I. Then q(U xj λi , U xj λi )(η) ≤ (ri + sj)(η) for all η ∈ A and i, j ∈ I. Thus max η q(Uxi λi , U xj λj )(η) ≤ ri + sj whenever i, j ∈ I. This implies that Mq(U xj λi , U xj λi ) ≤ ri + sj whenever i, j ∈ I. By Isbell convexity of (SP( ∼ U, Mq), we have ⋂ i∈I CMq (U xi λi , ri) ∩ CM qt (Uxi λi , si) 6= ∅. By Proposition 4.3, we have ⋂ i∈I Cq(U xi λi , ri) ∩ Cqt (U xi λi , si) = ⋂ i∈I CMq (U xi λi , ri) ∩ CM qt (Uxi λi , si) 6= ∅. Hence ∼⋂ i∈I SS(Cq(U xi λi , ri)) ∼ ∩ SS(Cqt(U xi λi , si)) 6= Φ. Therefore, ( ∼ U, q, A) is soft Isbell convex. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 25 H. Sabao and O. Olela Otafudu Definition 5.3. Let ( ∼ U, q, A) be a soft quasi-pseudometric space. A family of soft double balls [(SS(Cq(U xi λi , ri)))i∈I, (SS(Cqt(U xi λi , si)))i∈I], where ri and si are non-negative constant soft real numbers and U xi λi is a soft point of ∼ U whenever i ∈ I, is said to have a mixed binary intersection property if for all indices i, j ∈ I, SS(Cq(U xi λi , ri)) ∼ ∩ SS(Cqt (U xj λj , sj)) 6= Φ or equivalently, Cq(U xi λi , ri) ∩ Cqt(U xj λj , sj) 6= ∅. Definition 5.4. A soft quasi-pseudometric space ( ∼ U, q, A) is said to be soft Isbell complete if for every family of soft double balls [(SS(Cq(U xi λi , ri)))i∈I , (SS(Cqt(U xi λi , si)))i∈I], where ri and si are non-negative constant soft real numbers and U xi λi is a soft point of ∼ U whenever i ∈ I, having a mixed binary intersection property satisfy ∼⋂ i∈I SS(Cq(U xi λi , ri)) ∼ ∩ SS(Cqt (U xi λi , si)) 6= Φ or equivalently ⋂ i∈I Cq(U xi λi , ri) ∩ Cqt(U xi λi , si) 6= ∅. Lemma 5.5. A soft quasi-pseudometric space ( ∼ U, q, A), where A is finite, is soft Isbell complete if and only if (SP( ∼ U), Mq) is an Isbell complete quasi- pseudometric space. Proof. Suppose ( ∼ U, q, A) is soft Isbell complete. Let [(CMq (U xi λi , ri))i∈I, (CM qt (Uxi λi , si))i∈I], where ri and si are non-negative real numbers and U xi λi is a soft point of ∼ U whenever i ∈ I, have a mixed binary intersection property. Then CMq (U xi λi , ri) ∩ CM qt (U xj λj , sj) 6= ∅. By Proposition 4.3, Cq(U xi λi , ri) ∩ Cqt(U xj λj , sj) 6= ∅. Whenever i, j ∈ I. Hence the family of soft double balls [(SS(Cq(U xi λi , ri)))i∈I , (SS(Cqt(U xi λi , si)))i∈I], where ri and si are non-negative constant soft real numbers and U xi λi is a soft point of ∼ U whenever i ∈ I, have a mixed binary intersection property. By Isbell © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 26 On soft quasi-pseudometric spaces completeness of ( ∼ U, q, A), we have ∼⋂ i∈I SS(Cq(U xi λi , ri)) ∼ ∩ SS(Cqt (U xi λi , si)) 6= Φ or equivalently ⋂ i∈I Cq(U xi λi , ri) ∩ Cqt(U xi λi , si) 6= ∅. Therefore by Proposition 4.3, ⋂ i∈I CMq (U xi λi , ri) ∩ CM qt (Uxi λi , si) = ⋂ i∈I Cq(U xi λi , ri) ∩ Cqt(U xi λi , si) 6= ∅. Conversely, suppose (SP( ∼ U), q, A) is Isbell complete. Let [(SS(Cq(U xi λi , ri)))i∈I , (SS(Cqt(U xi λi , si)))i∈I], where ri and si are non-negative soft real numbers and U xi λi is a soft point of ∼ U whenever i ∈ I, be a family of soft double balls having a mixed binary intersection property. Then SS(Cq(U xi λi , ri)) ∼ ∩ SS(Cqt (U xj λj , sj)) 6= Φ or equivalently, Cq(U xi λi , ri) ∩ Cqt (U xj λj , sj) 6= ∅ whenever i, j ∈ I. By Proposition 4.3, we have CMq (U xi λi , ri) ∩ CM qt (U xj λj , sj) 6= ∅ whenever i, j ∈ I. Since (SP( ∼ U), q, A) is Isbell complete, it follows that ⋂ i∈I CMq (U xi λi , ri) ∩ CM qt (Uxi λi , si) 6= ∅. By Proposition 4.3, we have ⋂ i∈I Cq(U xi λi , ri) ∩ Cqt (U xi λi , si) = ⋂ i∈I CMq (U xi λi , ri) ∩ CM qt (Uxi λi , si) 6= ∅. Therefore, ∼⋂ i∈I SS(Cq(U xi λi , ri)) ∼ ∩ SS(Cqt(U xi λi , si)) 6= Φ. Hence, ( ∼ U, q, A) is soft Isbell complete. � Definition 5.6. A soft quasi-pseudometric space ( ∼ U, q, A) is said to be soft metrically convex if for any soft points Uxλ and U y µ of ∼ U and non-negative constant soft real numbers r and s, such that q(Uxλ , U y µ) ∼ ≤ r + s there exists a © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 27 H. Sabao and O. Olela Otafudu soft point Uzλ of ∼ U such that q(Uxλ , U z λ) ∼ ≤ r and q(Uzλ, U y µ) ∼ ≤ s or equivalently Cq(U x λ , r) ∩ Cqt(U y µ, s) 6= ∅. Remark 5.7. Notice that Cq(U x λ , r)∩Cqt (U y µ, s) 6= ∅ is equivalent to SS(Cq(U x λ , r))∩ SS(Cqt(U y µ, s)) 6= Φ. Lemma 5.8. A soft quasi-pseudometric space ( ∼ U, q, A), where A is finite, is soft metrically convex if and if (SP( ∼ U), Mq) is metrically convex. Proof. Suppose ( ∼ U, q, A) is soft metrically convex. Let Uxλ and U y µ be soft points of ∼ U and r and s are non-negative real numbers such that Mq(U x λ , U y µ) ≤ r + s. Then max η∈A q(Uxλ , U y µ)(η) ≤ r + s. Thus q(U x λ , U y µ)(η) ≤ r + s for all η ∈ A. Therefore, q(Uxλ , U y µ) ∼ ≤ r + s. By soft metric convexity of ( ∼ U, q, A), we have Cq(U x λ , r) ∩ Cqt (U y µ, s) 6= ∅. By Proposition 4.3, we have CMq (U x λ , r) ∩ CM qt (Uyµ, s) 6= ∅. Therefore, (SP( ∼ U), Mq) is metrically convex. Conversely, suppose (SP( ∼ U), Mq) is metrically convex. Let U x λ and U y µ be soft points of ∼ U and r and s are non-negative soft real numbers such that q(Uxλ , U y µ) ≤ r + s. Then q(U x λ , U y µ) ≤ r(η) + s(η) for all η ∈ A. Then max η∈A q(Uxλ , U y µ)(η) ≤ r + s and by soft metric convexity of (SP( ∼ U), Mq), we have CMq (U x λ , r)∩CMqt (U y µ, s) 6= ∅. Therefore, ( ∼ U, q, A) is soft metrically con- vex. � Lemma 5.9. A soft quasi-pseudometric space ( ∼ U, q, A), where A is finite, is soft Isbell convex if and if ( ∼ U, q, A) is soft-Isbell complete and soft metrically convex. Proof. Suppose ( ∼ U, q, A) is soft Isbell convex. Then by Lemma 5.2, (SP( ∼ U), Mq) is Isbell convex. By [13, Lemma 3.1.1], (SP( ∼ U), Mq) is Isbell complete and metrically convex. By Lemma 5.5 and Lemma 5.8, ( ∼ U, q, A) is soft metrically convex and soft Isbell complete. Conversely, suppose ( ∼ U, q, A) is soft metrically convex and soft Isbell complete. Then by Lemma 5.5 and Lemma 5.8, (SP( ∼ U), Mq) is Isbell complete and met- rically convex. Therefore, by [13, Lemma 3.1.1] (SP( ∼ U), Mq) is Isbell convex. Therefore, by Lemma 5.2, ( ∼ U, q, A) is soft metrically convex. � Proposition 5.10. Suppose ( ∼ U, q, A) is a soft Isbell convex soft quasi-metric space. Then ( ∼ U, q, A) is bicomplete. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 28 On soft quasi-pseudometric spaces Proof. Since ( ∼ U, q, A) is soft Isbell convex. Then (SP( ∼ U), Mq) is Isbell convex. By [13, Corollary 3.1.3] (SP( ∼ U), Mq) is bicomplete. This implies that ( ∼ U, q, A) is bicomplete by Theorem 4.1. � Definition 5.11. A soft quasi-pseudometric space ( ∼ U, q, A) is said to be bounded if for each soft points Uxλ and U y µ of ∼ U, there exists a positive soft real number k̂ such that q(Uxλ , U y µ) ∼ ≤ k̂. Remark 5.12. Notice that if ( ∼ U, q, A) is a soft quasi-pseudometric space, where A is finite, then boundedness of ( ∼ U, q, A) implies boundedness of (SP( ∼ U), Mq). Theorem 5.13. If ( ∼ U, q, A), where A is finite, is a bounded Isbell convex soft quasi-metric space and T : ( ∼ U, q, A) −→ ( ∼ U, q, A) is a non-expansive map, then the fixed point set Fix(T ) of T in ( ∼ U, q, A) is nonempty and soft Isbell convex. Proof. Since ( ∼ U, q, A) is soft Isbell convex, then (SP( ∼ U), Mq) is Isbell convex. Also, since ( ∼ U, q, A) is bounded, then (SP( ∼ U), Mq) is bounded by Remark 5.12. Furthermore, since f : ( ∼ U, q, A) −→ ( ∼ U, q, A) satisfies q(f(Uxλ ), f(U y µ)) ∼ ≤ q(Uxλ , U y µ) for all Uxλ , U y µ ∈ SP( ∼ U). 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