@ Appl. Gen. Topol. 18, no. 1 (2017), 203-217 doi:10.4995/agt.2017.6889 c© AGT, UPV, 2017 Metric spaces and textures Şenol Dost Hacettepe University, Department of Mathematics and Science Education, 06800 Beytepe, Ankara, Turkey. (dost@hacettepe.edu.tr) Communicated by S. Romaguera Abstract Textures are point-set setting for fuzzy sets, and they provide a frame- work for the complement-free mathematical concepts. Further dimetric on textures is a generalization of classical metric spaces. The aim of this paper is to give some properties of dimetric texture space by using categorical approach. We prove that the category of classical metric spaces is isomorphic to a full subcategory of dimetric texture spaces, and give a natural transformation from metric topologies to dimetric ditopologies. Further, it is presented a relation between dimetric tex- ture spaces and quasi-pseudo metric spaces in the sense of J. F. Kelly. 2010 MSC: 54E35; 54E40; 18B30; 54E15. Keywords: metric space; texture space; uniformity; natural transformation; difunction; isomorphism. 1. Introduction Texture theory is point-set setting for fuzzy sets and hence, some properties of fuzzy lattices (i.e. Hutton algebra) can be discussed based on textures [2, 3, 4, 5]. Ditopologies on textures unify the fuzzy topologies and classical topologies without the set complementation [6, 7]. Recent works on textures show that they are also useful model for rough set theory [8] and semi-separation axioms [10]. On the other hand, it was given various types of completeness for di- uniform texture spaces [13]. As an expanded of classical metric spaces, the dimetric notion on texture spaces was firstly defined in [11]. In this paper, we give the categorical properties of dimetric texture spaces, and present some relation between classical metric spaces and dimetric texture spaces. Received 20 November 2016 – Accepted 15 February 2017 http://dx.doi.org/10.4995/agt.2017.6889 Ş. Dost This section is devoted to some fundamental definitions and results of the texture theory from [2, 3, 4, 5, 6]. Definition 1.1. Let U be a set and U ⊆ P(U). Then U is called a texturing of U if (T1) ∅ ∈ U and U ∈ U, (T2) U is a complete and completely distributive lattice such that arbitrary meets coincide with intersections, and finite joins with unions, (T3) U is point-seperating. Then the pair (U,U) is called a texture space or texture. For u ∈ U, the p-sets and the q-sets are defined by Pu = ⋂ {A ∈ U | u ∈ A}, Qu = ∨ {A ∈ U | u /∈ A}, respectively. A texture (U,U) is said to be plain if Pu * Qu, ∀u ∈ U. A set A ∈ U\{∅} is called a molecule if A ⊆ B∪C, B,C ∈ U implies A ⊆ B or A ⊆ C. The texture (U,U) is called simple if the sets Pu, u ∈ U are the only molecules in U. Example 1.2. (1) For any set U, (U,P(U)) is the discrete texture with the usual set structure of U. Clearly, Pu = {u} and Qu = U \{u} for all u ∈ U, so (U,P(U)) is both plain and simple. (2) I = {[0, t] | t ∈ [0, 1]}∪{[0, t) | t ∈ [0, 1]} is a texturing on I = [0, 1]. Then (I,I) is said to be unit interval texture. For t ∈ I, Pt = [0, t] and Qt = [0, t). Clearly, (I,I) is plain but not simple since the sets Qu, 0 < u ≤ 1, are also molecules. (3) For textures (U,U) and (V,V), U⊗V is product texturing of U×V [5]. Note that the product texturing U⊗V of U ×V consists of arbitrary intersections of sets of the form (A×V )∪(U×B), A ∈ U and B ∈ V. Here, for (u,v) ∈ U×V P(u,v) = Pu ×Pv and Q(u,v) = (Qu ×V ) ∪ (U ×Qv). Ditopology: A pair (τ,κ) of subsets of U is called a ditopology on a texture (U,U) where the open sets family τ and the closed sets family κ satisfy U, ∅ ∈ τ, U, ∅ ∈ κ G1,G2 ∈ τ =⇒ G1 ∩G2 ∈ τ, K1, K2 ∈ κ =⇒ K1 ∪K2 ∈ κ Gi ∈ τ,i ∈ I =⇒ ∨ i∈I Gi ∈ τ, Ki ∈ κ,i ∈ I =⇒ ⋂ i∈I Ki ∈ κ. Direlation: Let (U,U), (V,V) be texture spaces. Now we consider the product texture P(U) ⊗ V of the texture spaces (U,P(U)) and (V,V). In this texture, p-sets and the q-sets are denoted by P (u,v) and Q(u,v), respectively. Clearly, c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 204 Metric spaces and textures P (u,v) = {u}× Pv and Q(u,v) = (U \ {u}× V ) ∪ (U × Qv) where u ∈ U and v ∈ V . According to: (1) r ∈ P(U) ⊗V is called a relation from (U,U) to (V,V) if it satisfies R1 r * Q(u,v),Pu′ * Qu =⇒ r * Q(u′,v). R2 r * Q(u,v) =⇒ ∃u′ ∈ U such that Pu * Qu′ and r * Q(u′,v). (2) R ∈ P(U) ⊗V is called a corelation from (U,U) to (V,V) if it satisfies CR1 P (u,v) * R,Pu * Qu′ =⇒ P (u′,v) * R. CR2 P (u,v) * R =⇒ ∃u′ ∈ U such that Pu′ * Qu and P (u′,v) * R. (3) If r is a relation and R is a corelation from (U,U) to (V,V) then the pair (r,R) is called a direlation from (U,U) to (V,V). A pair (i,I) is said to be identity direlation on (U,U) where i = ∨ {P (u,u) | u ∈ U} and I = ⋂ {Q(u,u) | U * Qu}. Recall that [5] we write (p,P) v (q,Q) if p ⊆ q and Q ⊆ P where (p,P) and (q,Q) are direlations. Let (p,P) and (q,Q) be direlations from (U,U)to (V,V). Then the greatest lower bound of (p,P) and (q,Q) is denoted by (p,P)u(q,Q) , and it is defined by (p,P) u (q,Q) = (puq,P tQ) where puq = ∨ {P (u,v) | ∃z ∈ U with Pu * Qz, and p,q * Q(z,v)}, P tQ = ⋂ {Q(u,v) | ∃z ∈ U with Pz * Qu, and P (z,v) * P,Q}. Inverses of a direlation: If (r,R) is a direlation then the inverse direlation of (r,R)← is a direlation from (V,V) to (U,U), and it is defined by (r,R)← = (R←,r←) where r← = ⋂ {Q(v,u) | r * Q(u,v)} and R ← = ∨ {P (v,u) | P (u,v) * R} The A-sections and the B-presections under a direlation (r,R) are defined as r→A = ⋂ {Qv | ∀u,r * Q(u,v) =⇒ A ⊆ Qu}, R→A = ∨ {Pv | ∀u,P (u,v) * R =⇒ Pu ⊆ A}, r←B = ∨ {Pu | ∀v,r * Q(u,v) =⇒ Pv ⊆ B}, R←B = ⋂ {Qu | ∀v,P (u,v) * R =⇒ B ⊆ Qv}. The composition of direlations: Let (p,P) be a direlation from (U,U) to (V,V), and (q,Q) be a direlation on (V,V) to (W,W). The composition c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 205 Ş. Dost (q,Q) ◦ (p,P) of (p,P) and (q,Q) is a direlation from (U,U) to (W,W) and it is defined by (q,Q) ◦ (p,P) = (q ◦p,Q◦P) where q ◦p = ∨ {P (u,w) | ∃v ∈ V with p * Q(u,v) and q * Q(v,w)}, Q◦P = ⋂ {Q(u,w) | ∃v ∈ V with P (u,v) * P and P (v,w) * Q}. Difunction: A direlation from (U,U) to (V,V) is called a difunction if it satisfies the conditions: (DF1) For u,u′ ∈ U, Pu * Qu′ =⇒ ∃v ∈ V with f * Q(u,v) and P (u′,v) * F . (DF2) For v,v′ ∈ V and u ∈ U, f * Q(u,v) and P (u,v′) * F =⇒ Pv′ * Qv. Obviously, identity direlation (i,I) on (U,U) is a difunction and it is said to be identity difunction. It is well known that [5] the category dfTex of textures and difunctions is the main category of texture theory. Definition 1.3. Let (f,F) : (U,U) → (V,V) be a difunction. If (f,F) satisfies the condition SUR. For v,v′ ∈ V , Pv * Qv′ =⇒ ∃u ∈ U with f * Q(u,v′) and P (u,v) * F . then it is called surjective. Similarly, (f,F) satisfies the condition INJ. For u,u′ ∈ U and v ∈ V , f * Q(u,v) and P (u′,v) * F =⇒ Pu * Qu′ . then it is called injective. If (f,F) is both injective and surjective then it is called bijective. Note 1.4. In general, difunctions are not directly related to ordinary (point) functions between the base sets. We note that [5, Lemma 3.4] if (U,U), (V,V) are textures and a point function ϕ : U → V satisfies the condition (a) Pu * Qu′ =⇒ Pϕ(u) * Qϕ(u′) then the equalities fϕ = ∨ {P (u,v) | ∃z ∈ U satisfying Pu * Qz and Pϕ(z) * Qv}, Fϕ = ⋂ {Q(u,v) | ∃z ∈ U satisfying Pz * Qu and Pv * Qϕ(z)}, define a difunction (fϕ,Fϕ) on (U,U) to (V,V). For B ∈ V, F←ϕ B = ϕ←B = f←ϕ B, where ϕ ←B = ∨ {Pu | Pϕ(u′) ⊆ B ∀u′ ∈ U with Pu * Qu′}. Furthermore, the function ϕ = ϕ(f,F) : U → V corresponding as above to the difunction (f,F) : (U,U) → (V,V), with (V,V) plain, has the property (a) and in addition the property: (b) Pϕ(u) * B, B ∈ V =⇒ ∃u′ ∈ U with Pu * Qu′ for which Pϕ(u′) * B. Conversely, if ϕ : U → V is any function satisfying (a) and (b) then there exists a unique difunction (fϕ,Fϕ) : (U,U) → (V,V) satisfying ϕ = ϕ(fϕ,Fϕ). c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 206 Metric spaces and textures On the other hand, if we consider simple textures it is obtained the same class of point functions. The category of textures and point functions which satisfy the conditions (a)- (b) between the base sets is denoted by fTex. Bicontinuous Difunction: A difunction (f,F) : (U,U,τU,κU ) → (V,V,τV ,κV ) is called continuous (cocontinuous) if B ∈ τV (B ∈ κV ) =⇒ F←(B) ∈ τU (f←(B) ∈ κU ). A difunction (f,F) is called bicontinuous if it is both continuous and co- continuous. The category of ditopological texture spaces and bicontinuous difunctions was denoted by dfDitop in [6]. 2. Some categories of dimetrics on texture spaces The notion of dimetric on texture space was firstly introduced in [11]. In this section, we will give some properties of dimetric texture spaces, and we present a link between classical metrics and dimetrics with categorical approach. Definition 2.1. Let (U,U) be a texture, ρ,ρ : U × U → [0,∞) two point function. Then ρ = (ρ,ρ) is called a pseudo dimetric on (U,U) if M1 ρ(u,z) ≤ ρ(u,v) + ρ(v,z), M2 Pu * Qv =⇒ ρ(u,v) = 0, DM ρ(u,v) = ρ(v,u), CM1 ρ(u,z) ≤ ρ(u,v) + ρ(v,z), CM2 Pv * Qu =⇒ ρ(u,v) = 0. for all u,v,z ∈ U. In this case ρ is called pseudo metric, ρ the pseudo cometric of ρ. If ρ is a pseudo dimetric which satisfies the conditions M3 Pu * Qv,ρ(v,y) = 0,Py * Qz =⇒ Pu * Qz ∀u,v,y,z ∈ U, CM3 Pv * Qu,ρ(u,y) = 0,Pz * Qy =⇒ Pz * Qu ∀u,v,y,z ∈ U it is called a dimetric. If ρ = (ρ,ρ) is (pseudo) dimetric on (U,U) then (U,U,ρ) is called (pseudo) dimetric texture space. Let (U,U,ρ) be a (pseudo) dimetric texture space. It was shown in [11, Proposition 6.3] that βρ = {Nρ� (u) | u ∈ U[,� > 0} is a base and γρ = {Mρ� (u) | u ∈ U[,� > 0} a cobase for a ditopology (τρ,κρ) on (U,U) where Nρ� (u) = ∨ {Pz | ∃v ∈ U, with, Pu * Qv,ρ(v,z) < �}, Mρ� (u) = ⋂ {Qz | ∃v ∈ U, with, Pv * Qu,ρ(v,z) < �}. In this case (U,U,τρ,κρ) is said to be (pseudo) dimetric ditopological texture space. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 207 Ş. Dost Definition 2.2. Let (U,U,ρ) be a (pseudo) dimetric texture space. Then G ∈ U is called (1) open if for every G * Qu, then there exists � > 0 such that Nρ� (u) ⊆ G, (2) closed if for every Pu * G, then there exists � > 0 such that G ⊆ Mρ� (u). We set Oρ = {G ∈ U | G is open in (U,U,ρ)} and Cρ = {K ∈ U | K is closed in (U,U,ρ)}. Proposition 2.3. Let (U,U,ρ) be a (pseudo) dimetric texture space. For u ∈ U and � > 0, (i) Nρ� (u) is open in (U,U,ρ), (ii) Mρ� (u) is closed in (U,U,ρ). Proof. We prove (i), and the second result is dual. Let Nρ� (u) * Qv for some v ∈ U. By the definition of Nρ� (u), there exists y,z ∈ U such that Py * Qv and Pu * Qz, ρ(z,y) < �. We set δ = �−ρ(z,y). Now we show that N ρ δ (v) ⊆ N ρ � (u). We suppose N ρ δ (v) * N ρ � (u). Then N ρ δ (v) * Qr and Pr * N ρ � (u) for some r ∈ U. By the first inclusion, there exists m,n ∈ U such that Pm * Qr, Pv * Qn and ρ(n,m) < δ. Now we observe that ρ(z,y) + ρ(n,m) < � and ρ(z,r) ≤ ρ(z,y) + ρ(y,v) + ρ(v,n) + ρ(n,m) ≤ � by the condition (M2). Since Pu * Qz and ρ(z,r) ≤ �, we have the contradic- tion Pr ⊆ Nρ� (u). � Definition 2.4. Let (Uj,Uj,ρj), j = 1, 2 be (pseudo) dimetric texture spaces and (f,F) be a difunction from (U1,U1) to (U2,U2). Then (f,F) is called (1) ρ1 −ρ2 continuous if P (u,v) * F then N ρ1 δ (u) ⊆ F ←(Nρ2� (v)), ∀� > 0 ∃δ > 0, (2) ρ1 −ρ2 cocontinuous if f * Q(u,v) then f←(Mρ2� (v)) ⊆ M ρ1 δ (u), ∀� > 0 ∃δ > 0, (3) ρ1 −ρ2 bicontinuous if it is continuous and cocontinuous. Proposition 2.5. Let (f,F) be a difunction from (U1,U1,ρ1) to (U2,U2,ρ2). (i) (f,F) is continuous ⇐⇒ F←(G) ∈ Oρ1 , ∀G ∈ Oρ2 . (ii) (f,F) is cocontinuous ⇐⇒ f←(K) ∈ Cρ1 , ∀K ∈ Cρ2 . Proof. We prove (i), and the second result is dual. (=⇒:) Let (f,F) be a continuous difunction. Take G ∈ Oρ2 . We show that F←(G) is open in (U1,U1,ρ1). Let F ←(G) * Qu for some u ∈ U. By the definition of inverse image, there exists v ∈ V such that P (u,v) * F and G * Qv. Since G is open, we have Nρ� (v) ⊆ G for � > 0. By the definition of continuity, there exists δ > 0 such that N ρ1 δ (u) ⊆ F ←(Nρ2� (v)). Then N ρ1 δ (u) ⊆ F ←(G), and so F←(G) ∈ Oρ1 . (⇐=:) Let P (u,v) * F. We consider Nρ2� (v) for some � > 0. Since Nρ2� (v) ∈ Oρ2 , we have F ←(Nρ2� (v)) ∈ Oρ1 by assumption. Since Pv ⊆ Nρ2� (v) and Pv * F→(Qu), Nρ2� (v) * F →(Qv). Hence, we have F ←(Nρ2� (v)) * Qu. Then there exists δ > 0 such that N ρ1 δ (u) ⊆ F ←(Nρ2� (v)). � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 208 Metric spaces and textures Corollary 2.6. Let (Uj,Uj,ρj), j = 1, 2 be (pseudo) dimetric texture spaces and (f,F) be a difunction from (U1,U1) to (U2,U2). (1) (f,F) is ρ1 −ρ2 continuous ⇐⇒ (f,F) is τρ1 − τρ2 continuous. (2) (f,F) is ρ1 −ρ2 cocontinuous ⇐⇒ (f,F) is κρ1 −κρ2 cocontinuous. Proof. We prove (1), leaving the dual proof of (2) to the interested reader. (=⇒:) Let (f,F) be ρ1 −ρ2 continuous. Let G ∈ τρ2 . To prove F←(G) ∈ τρ1 , we take F←(G) * Qu for some u ∈ U1. By definition of inverse relation, there exists v ∈ U2 such that P (u,v) * F and G * Qv. Since G ∈ τρ2 , we have Nρ2� (v) ⊆ G for some � > 0. Then F← ( Nρ2� (v) ) ⊆ F←(G). From the assumption, we have δ > 0 such that N ρ1 δ (u) ⊆ F ← ( Nρ2� (v) ) ⊆ F←(G). Thus, F←(G) ∈ τρ1 . (⇐=:) Let P (u,v) * F. We consider Nρ2� (v) for some � > 0. Since Pv ⊆ Nρ2� (v), we have Nρ2� (v) * F →(Qu). Hence, F ← ( Nρ2� (v) ) * Qu, and since F← ( Nρ2� (v) ) is open in (U1,U1,ρ1), we have N ρ1 � (u) ⊆ F← ( N ρ2 δ (v) ) for some δ > 0. Thus, (f,F) is ρ1 −ρ2 continuous. � Theorem 2.7. (Pseudo) dimetric texture spaces and bicontinuous difunctions form a category. Proof. Since bicontinuity between ditopological texture spaces is preserved un- der composition of difunction [6], and identity difunction on (S,S,ρ) is ρ − ρ bicontinuous, and the identity difunctions are identities for composition and composition is associative [5, Proposition 2.17(3)], (pseudo) di-metric texture spaces and bicontinuous difunctions form a category. � Definition 2.8. The category whose objects are (pseudo) di-metrics texture spaces and whose morphisms are bicontinuous difunctions will be denoted by (dfDiMP) dfDiM. Clearly, dfDiM is a full subcategory of dfDiMP. If we take as objects di-metric on a simple texture we obtain the full sub- category dfSDiM and inclusion functor S : dfSDiM ↪→ dfDiM. Also we obtain the full subcategory dfPDiM and inclusion functor P : dfPDiM ↪→ dfDiM by taking as objects di-metrics on a plain texture. In the same way we can use dfPSDiM to denote the category whose objects are di-metrics on a plain simple texture, and whose morphisms are bicontinuous difunctions. Now, we define G : dfDiM → dfDitop by G((U,U,ρ) (f,F) −−−→ (V,V,µ)) = (U,U,τρ,κρ) (f,F) −−−→ (V,V,τµ,κµ). Obviously, G is a full concrete functor from Corollary 2.6. Likewise, the same functor may set up from dfDiMP to dfDitop. We recall [11] that a ditopology on (U,U) is called (pseudo) dimetrizable if it is the (pseudo) dimetric ditopology of some (pseudo) dimetric on (U,U). We c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 209 Ş. Dost denote by dfDitopdm the category of dimetrizable ditopological texture space and bicontinuous difunction. Clearly it is full subcategory of the category dfDitop. Proposition 2.9. The categories dfDitopdm and dfDiM are equivalent. Proof. Consider the functor G : dfDiM → dfDitopdm which is defined above. It can be easily seen that G is a full and faithfull functor, since the hom- set restriction function of G is onto and injective. Now we take a dimetrizable ditopological texture space (U,U,τ,κ) such that τ = τρ and κ = κρ, where ρ is a dimetric on (U,U). Clearly, the identity difunction (i,I) : (U,U,ρ) → G(U,U,ρ) is an isomorphism in the category dfDitopdm. Hence, G is isomorphism-closed, and so the proof is completed. � Corollary 2.10. The category dfDiMP is equivalent to the category of pseudo dimetrizable completely biregular [7] ditopological texture spaces and bicontinu- ous difunctions. Proof. Let (U,U,ρ) be a pseudo dimetric space. Then the dimetric ditopology (U,U,τρ,κρ) is completely biregular by [11, Corollary 6.5]. Consequently, the functor G which is the above proposition is given an equivalence between the categories dfDiMP and the category of pseudo metrizable completely biregular ditopological texture spaces. � On the other hand, since every pseudo dimetric ditopology is T0 [11, Corol- lary 6.5], so the category dfDiMP is equivalent to the category of pseudo metrizable T0 ditopological texture spaces and bicontinuous difunctions. Now we give some properties of morphisms in the category dfDiM. Note that it takes consideration the reference [1] for some concepts of category theory Proposition 2.11. Let (f,F) be a morphism from (U,U,ρ) to (V,V,µ) in the category dfDiM (dfDiMP). (1) If (f,F) is a section then it is injective. (2) If (f,F) is injective morphism then it is a monomorphism. (3) If (f,F) is retraction then it is surjective. (4) If (f,F) is surjective morphism then it is an epimorphism. (5) (f,F) is an isomorphism if and only if it is bijective and the inverse difunction (f,F)← is bicontinuous difunction. Proof. The proof of (1)−(4) can be obtained easily in the category dfTex by [5, Proposition 3.14]. We show that the result (5). Note that, (f,F) is bijective if and only if it is an isomorphism in dfTex. Since (f,F) is bijective, its inverse (f,F)← is a morphism in dfTex such that (f,F)← ◦ (f,F) = (iU,IU ), (f,F) ◦ (f,F)← = (iV ,IV ). Consequently, (f,F) is ρ − µ bicontinuous iff (f,F)← is µ−ρ bicontinuous. � Now let (U,d) be a classical (pseudo) metric space. Then ρ = (d,d) is a (pseudo) dimetric on the discrete texture space ( U,P(U) ) . As a result, a c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 210 Metric spaces and textures subset of U is open (closed) in the metric space (U,d) if and only if it is open (closed) in the dimetric texture space (U,P(U),ρ). On the other hand, recall that [5] if (f,F) is a difunction from (U,P(U)) to (V,P(V )), then f and F are point functions from U to V where F = (U ×V )\ f = f′. The category of metric spaces and continuous functions between metric spaces is denoted by Met. According to: Theorem 2.12. The category Met is isomorphic to the full subcategory of dfDiM. Proof. We consider a full subcategory D-dfDiM of dfDiM whose objects are dimetric texture spaces on discrete textures and morphisms are bicontinuous difunctions. Now we prove that the mapping T :Met→ D-dfDiM is a functor such that T(U,d) = (U,P(U),ρ) and T(f) = (f,f′) where f is a morphism in Met. Note that (f,f′) is a bicontinuous difunction in D-dfDiM if and only if f is a continuous point function in Met. It can be easily seen that if i is identity function on U then (i,I) is identity difunction on (U,P(U)) where I = (U ×U)\ i. Since f′◦g′ = (f ◦g)′, we have T(f ◦g) = T(f)◦T(g). Hence, T is a functor. Furthermore, T is bijective on objects, and the hom-set restriction of T is injective and onto. Consequently, T is clearly an isomorphism functor. � By using same arguments, the category PMet of pseudo metric spaces and continuous functions is isomorphic to the full subcategory of dfDiMP. Now suppose that (U,d) is a classical metric space and (U,Td) is the metric topological space. Then the pair (Td,T c d) is a ditopology on (U,P(U)). On the other hand, we consider the dimetric ditopological texture space (U,P(U),τρ,κρ) where ρ = (d,d). Now we consider the functors M : Met → dfDitop and N : Met → dfDitop which are defined by M((U,d) f−→ (V,e)) = (U,P(U),Td,Tcd) (f,f′) −−−→ (V,P(V ),Te,Tce), N((U,d) f−→ (V,e)) = (U,P(U),τρ,κρ) (f,f′) −−−→ (V,P(V ),τµ,κµ) where ρ = (d,d) and µ = (e,e). According to: Proposition 2.13. Let τ : M → N be a function such that assigns to each Met-object (X,d) a dfDitop-morphism τ(X,d) = (i,I) : M(X,d) → N(X,d). Then τ is a natural transformation. Proof. We prove that naturality condition holds. Let f : (U,d) → (V,e) be a Met-morphism. From Theorem 2.12, (f,f′) : (U,P(U),ρ) → (V,P(V ),µ) is a dfDiM-morphism. Further, it is a dfDitop-morphism by Corollary 2.6. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 211 Ş. Dost (U,P(U),Td,T c d) τ(U,d)=(i,I) // (f,f′) �� (U,P(U),τρ,κρ) (f,f′) �� (V,P(V ),Te,T c e) τ(V,e)=(i,I) // (V,P(V ),τµ,κµ) On the other hand, the identity difunction τ(U,d) = (i,I) : (U,P(U),Td,T c d) → (U,P(U),τρ,κρ) is bicontinuous on (U,P(U)), and so it is a dfDitop-morphism. Clearly the above diagram is commutative, and the proof is completed. � 3. Point functions between dimetric texture spaces As we have noted earlier, however, it is possible to represent difunctions by ordinary point functions in certain situations. The construct fDitop, where the objects are ditopological texture spaces and the morphisms bicontinuous point functions satisfying (a) and (b) which is given Note 1.4, and we will to define a similar construct of (pseudo) dimetric texture spaces. Definition 3.1. Let (U,U,ρ) and (V,V,µ) be (pseudo) dimetric texture spaces, and ϕ on U to V a point function satisfy the condition (a). Then ϕ is called (1) continuous if ϕ←(Pv) * Qu implies N ρ δ (u) ⊆ ϕ ←(Nµ� (v)), ∀� > 0 ∃δ > 0. (2) cocontinuous if Pu * ϕ←(Qv) implies ϕ←(Mµ� (v)) ⊆ M ρ δ (u), ∀� > 0 ∃δ > 0. (3) bicontinuous if it is continuous and cocontinuous. Proposition 3.2. Let ϕ be a point function satisfy the condition (a) from (U1,U1,ρ1) to (U2,U2,ρ2). (i) ϕ is continuous ⇐⇒ ϕ←(G) ∈ Oρ1 , ∀G ∈ Oρ2 . (ii) ϕ is cocontinuous ⇐⇒ ϕ←(K) ∈ Cρ1 , ∀K ∈ Cρ2 . Proof. Let ϕ be a point function satisfy the condition (a) and (fϕ,Fϕ) be the corresponding difunction. Then ϕ←(B) = F←ϕ (B) = f ← ϕ (B) for all B ∈ U2. Now we take G ∈ Oρ2 . We show that ϕ←(G) is open in (U1,U1,ρ1). Let ϕ←(G) * Qu for some u ∈ U. By the definition of inverse image, there exists v ∈ V such that P (u,v) * F and G * Qv. Since G is open, we have Nρ2� (v) ⊆ G for � > 0. By the definition of continuity, there exists δ > 0 such that N ρ1 δ (u) ⊆ F←(Nρ2� (v)). Then N ρ1 δ (u) ⊆ F ←(G), and so F←(G) ∈ Oρ1 . (⇐=:) Let P (u,v) * F. We consider Nρ2� (v) for some � > 0. Since Nρ2� (v) ∈ Oρ2 , we have F ←(Nρ2� (v)) ∈ Oρ1 by assumption. Since Pv ⊆ Nρ2� (v) and Pv * F→(Qu), Nρ2� (v) * F →(Qv). Hence, we have F ←(Nρ2� (v)) * Qu. Then there exists δ > 0 such that N ρ1 δ (u) ⊆ F ←(Nρ2� (v)). � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 212 Metric spaces and textures Corollary 3.3. Suppose that ϕ : (U1,U1) → (U2,U2) is a point function satisfy the condition (a) and that ρk is a (pseudo) dimetric on (Uk,Uk), k = 1, 2. Then (1) ϕ is bicontinuous if and only if (fϕ,Fϕ) is bicontinuous. (2) ϕ is ρ1 − ρ2 bicontinuous if and only if ϕ is (τρ1,κρ1 )–(τρ2,κρ2 ) bi- continuous where (τρj,κρj ), j = 1, 2 is dimetric ditopological texture space. Proof. Since ϕ←(B) = F←ϕ (B) = f ← ϕ (B) for all B ∈ U2, the proof is automati- cally obtained by Corollary 2.6. � The category whose objects are dimetrics and whose morphisms are bicon- tinuous point functions satisfying the conditions (a) and (b) will be denoted by fDiM. Proposition 3.4. Let f be a morphism from (U,U,ρ) to (V,V,µ) in the cate- gory fDiM. (1) If f is a section then it is an fDiM-embedding. (2) If f is injective morphism then it is a monomorphism. (3) If f is a retraction then it is a fDiM-quotient. (4) If f is a surjective morphism then it is an epimorphism. (5) f is an isomorphism if and only if it is a textural isomorphism and its inverse is bicontinuous. Proof. Since the category fDiM is a construct, the first four results are auto- matically obtained. Recall that f is a textural isomorphism from (U,U) to (V,V) if it is a bijective point function from U to V satisfying A ∈ U =⇒ f(A) ∈ V such that A → f(A) is a bijective from U to V. Hence, this is equivalent to requiring that f be bijective with inverse g, and A ∈ U =⇒ f(A) ∈ V and B ∈ V =⇒ g(B) ∈ U. By [5, Proposition 3.15], f is textural isomorphism if and only if f is isomor- phism in fTex. � We define D : fDiM → dfDiM by D((U,U,ρ) ϕ−→ (V,V,µ)) = (U,U,ρ) (fϕ,Fϕ)−−−−−→ (V,V,µ). Theorem 3.5. D : fDiM → dfDiM defined above is a functor. The re- striction Dp : fPDiM → dfPDiM is an isomorphism with inverse Vp : dfPDiM → fPDiM given by Vp((U,U,ρ) (f,F) −−−→ (V,V,µ)) = (U,U,ρ) ϕ(f,F )−−−−→ (V,V,µ). Likewise we have isomorphism between fSDiM and dfSDiM. Proof. It is easy to show that D(ιU ) = (iU,IU ). Now let (U,U), (V,V), (Z,Z) be textures, ϕ : U → V , ψ : V → Z point functions satisfying (a) and (b). We have (fψ◦ϕ,Fψ◦ϕ) = (fψ,Fψ) ◦ (fϕ,Fϕ) by [5, Theorem 3.10]. We can also say that a point function is (texturally) bicontinuous if and only if the corresponding c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 213 Ş. Dost difunction is bicontinuous. Thus D : fDiM → dfDiM is a functor. If we restrict to Dp : fPDiM → dfPDiM we again obtain a functor. Now let us define Vp : dfPDiM → fPDiM by Vp(U,U,ρ) = (U,U,ρ) and Vp(f,F) = ϕ(f,F) which is also a functor and the inverse of Dp. This means that Dp is an isomorphism. The other isomorphisms can be proved similarly. � We recall that a quasi-pseudo metric on a set U in the sense of J. C. Kelly [9] is a non-negative real-valued function ρ(, ) on the product U ×U such that (1) ρ(u,u) = 0, (u ∈ U) (2) ρ(u,z) ≤ ρ(u,v) + ρ(v,z), (u,v,z ∈ U) Now let ρ(, ) be a quasi-pseudo metric on a set U, and let q(, ) be defined by q(u,v) = ρ(v,u). Then it is a trivial matter to verify that q(u,v) is a quasi- pseudo metric on U. In this case, ρ(, ) and q(, ) are called conjugate, and denote the set U with this structure (U,ρ,q). Now let (U1,ρ1,q1) and (U2,ρ2,q2) be quasi-pseudo metric spaces. A function f : U1 → U2 is pairwise continuous if and only if f is ρ1–ρ2 continuous and q1–q2 continuous. So, quasi-pseudo metric spaces and pairwise continuous functions form a category, and we will denote this category PQMet. Obviously, Met is a full subcategory of PQMet. Now let (U,U,ρ) be a dimetric space with (U,U) plain. Then u = v =⇒ ρ(u,v) = 0 and ρ(u,v) = 0, by the dimetric condition (M2). So, (U,ρ,ρ) is pseudo-quasi metric space in the usual sense. Thus we have a forgetful functor A : fPSDiMP → PQMet, if we set A(U,U,ρ) = (U,ρ,ρ) and A(ϕ) = ϕ. Likewise, the functor T : Met → dfDiM becomes a functor T : PQMet → dfDiMP on setting T(U,p,q) = (U,P(U), (p,q)) and T(ϕ) = ϕ. Now we consider the following diagram. PQMet T uukkk kkk kkk k dfPSDiMP Vps // fPSDiMP A iiRRRRRRRRR Then: Theorem 3.6. A is an adjoint of Vps ◦T and T a co-adjoint of A◦Vps. Proof. Take (U,p,q) ∈ Ob (PQMet). We show that (ιU, (U,P(U), (p,q))) is an A-universal arrow. It is clearly an A-structured arrow, so take an object (U,U,µ) in fPSDiMP and ϕ ∈ PQMet((U,p,q), (U,µ,µ)). Then, by [5, Theorem 3.12], we know that ϕ ∈ Mor fPSTex, and that it is the unique such morphism satisfying A(ϕ) ◦ ιU = ϕ, so it remains to verify that ϕ : (U,P(U), (p,q)) → (U,U,µ) is bicontinuous. However, for every open set G in (U,U,µ), we have ϕ←(G) = ϕ−1[G], by [5, Lemma 3.9], and ϕ−1[G] is open in (U,p,q) since ϕ is p–µ continuous. Likewise, for every closed set K in (U,U,µ) we have ϕ←(K) = U \ϕ−1[U \K] is closed in (U,p,q) since ϕ is q–µ continuous. � c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 214 Metric spaces and textures 4. Dimetrics and Direlational Uniformity In this section, we will give a relation between dimetrics and direlational uni- formity by using categorical approach. Firstly, we recall some basic definitons and results for direlational uniformity from [11]. Let us denote by DR the family of direlations on (U,U). Direlational Uniformity: Let (U,U) be a texture space and D a family of direlations on (U,U). Then D is called direlational uniformity on (U,U if it satisfies the following conditions: (1) If (r,R) ∈ D implies (i,I) v (r,R). (2) If (r,R) ∈ D, (e,E) ∈ DR and (r,R) v (e,E) then (e,E) ∈ D. (3) If (r,R), (e,E) ∈ D implies (d,D) u (e,E) ∈ D. (4) If (r,R) ∈ D then there exists (e,E) ∈ D such that (e,E) ◦ (e,E) v (r,R). (5) If (r,R) ∈ D then there exists (c,C) ∈ U such that (c,C)← v (r,R). Then the triple (U,U,D) is said to be direlational uniform texture. It will be noted that this definition is formally the same as the the usual definition of a diagonal uniformity, and the notions of base and subbase may be defined in the obvious way. Further, if d D = (i,I) then D is said to be separated. Inverse of a direlation under a difunction: Let (f,F) be a difunction from (U,U) to (V,V) and (r,R) be a direlation on (V,V). Then (f,F)−1(r) = ∨ {P (u1,u2) | ∃Pu1 * Qu′1 so that P (u′1,v1) * F,f * Q(u2,v2) =⇒ P (v1,v2) ⊆ r} (f,F)−1(R) = ⋂ {Q(u1,u2) | ∃Pu′1 * Qu1 so thatf * Q(u′1,v1),P (u2,v2) * F, =⇒ R ⊆ Q(v1,v2)} (f,F)−1(r,R) = ((f,F)−1(r), (f,F)−1(R)). Uniformly bicontinuos difunction: Let (U,U,D) and (V,V,E) be direla- tional uniform texture space and (f,F) be a difunction from (U,U) to (V,V). Then (f,F) is called D–E uniformly bicontinuous if (r,R) ∈ E =⇒ (f,F)−1(r,R) ∈ D. Recall that [12] the category whose objects are direlational uniformities and whose morphisms are uniformly bicontinuous difunctions was denoted by dfDiU. Now let us verify that a pseudo dimetric also defines a direlational uniformity. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 215 Ş. Dost Theorem 4.1. Let ρ be a pseudo dimetric on (U,U). i) For � > 0 let r� = ∨ {P (u,v) | ∃z ∈ U,Pu * Qz and ρ(z,v) < �} R� = ⋂ {Q(u,v) | ∃z ∈ U,Pz * Qu and ρ(z,v) < �} Then the family {(r�,R�) | � > 0} is a base for a direlational uniformity Dρ on (U,U). ii) The uniform ditopology [11] of Uρ coincides with the pseudo metric ditopology of ρ. A direlational uniformity D on (U,U) is called (pseudo) dimetrizable if there exists a (pseudo) dimetric ρ with D = Dρ. Lemma 4.2. Let (Uj,Uj,ρj), j = 1, 2 be (pseudo) dimetrics and (f,F) be a difunction from (U1,U1) to (U2,U2). Then (f,F) is ρ1−ρ2 bicontinuous if and only if (f,F) is Dρ1 −Dρ2 uniformly bicontinuous. Proof. Let (f,F) be a ρ1 − ρ2 bicontinuous difunction from (U1,U1,ρ1) to (U2,U2,ρ2). From Corollary 2.6, (f,F) is also bicontinuous from (U1,U1,τρ1,κρ1 ) to (U2,U2,τρ2,κρ2 ) where (τρj,κρj ) is (pseudo) dimetric ditopology on (Uj,Uj), j = 1, 2. On the other hand, the uniform ditopology of Dρj coincides with the (pseudo) dimetric ditopology of ρj, j = 1, 2. Further, (f,F) is also uniformly bicontinuous by [11, Proposition 5.13]. � Now we define G : dfDiM → dfDiU by G((U,U,ρ) (f,F) −−−→ (V,V,µ)) = (U,U,Dρ) (f,F) −−−→ (V,V,Dµ). Obviously, G is a full concrete functor from Lemma 4.2. We denote by dfDiUdm the category of dimetrizable direlational uniform tex- tures and uniformly bicontinuous difunctions. Proposition 4.3. The categories dfDiUdm and dfDiM are equivalent. Proof. It is easy to show that the functor G : dfDiM → dfDiUdm which is defined above is full and faitfull. Now we take an object (U,U,D) in dfDiUdm. Since it is a metrizable direlational uniform space, then there exists a dimetric ρ on (U,U) such that U = Uρ. Because of the identity difunction (i,I) : (U,U,ρ) → G(U,U,ρ) is an isomorphism in the category dfDiUdm, the functor G is isomorphism-closed. Hence, the proof is completed. � Recall that [11] a direlational uniformity U is (pseudo) dimetrizable if and only if it has a countable base. If the category of direlational uniformities with countable bases and uniformly bicontinuous difunctions denote by dfDiUcb then we have next result automatically from Proposition 4.3: Corollary 4.4. The categories dfDiUcb and dfDiM are equivalent. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 216 Metric spaces and textures A direlational uniformity D is dimetrizable if and only if it is separated [11]. We denote the category of separated direlational uniformities and uniformly bicontinuous difunctions by dfDiUs. From Proposition 4.3, we have: Corollary 4.5. dfDiUs is equivalent to the category dfDiM. Acknowledgements. The author would like to thank the referees and ed- itors for their helpful comments that have helped improve the presentation of this paper. References [1] J. Adámek, H. Herrlich and G. E. Strecker, Abstract and concrete categories, John Wiley & Sons, Inc., 1990. [2] L. M. Brown and M. Diker, Ditopological texture spaces and intuitionistic sets, Fuzzy Sets and Systems 98 (1998), 217–224. [3] L. M. Brown and R. Ertürk, Fuzzy sets as texture spaces, I. Representation Theorems, Fuzzy Sets and Systems 110, no. 2 (2000), 227–236. [4] L. M. Brown and R. Ertürk, Fuzzy sets as texture spaces, II. Subtextures and quotient textures, Fuzzy Sets and Systems 110, no. 2 (2000), 237–245. [5] L. M. Brown, R. Ertürk and Ş. Dost, Ditopological texture spaces and fuzzy topology, I. Basic Concepts, Fuzzy Sets and Systems 147, no. 2 (2004), 171–199. [6] L. M. Brown, R. Ertürk and Ş. Dost, Ditopological texture spaces and fuzzy topology, II. Topological Considerations, Fuzzy Sets and Systems 147, no. 2 (2004), 201–231. [7] L. M. Brown, R. Ertürk and Ş. Dost, Ditopological texture spaces and fuzzy topology, III. Separation Axioms, Fuzzy Sets and Systems 157, no. 14 (2006), 1886–1912. [8] M. Diker and A. Altay Uğur, Textures and covering based rough sets, Information Sciences 184 (2012), 33–44. [9] J. L. Kelley, General topology, D. Van Nostrand, Princeton, 1995. [10] M. Kule and Ş. Dost, A textural view of semi-separation axioms in soft fuzzy topological spaces, Journal of Intelligent Fuzzy Systems 32 (2017), 925–936. [11] S. Özçağ and L. M. Brown, Di-uniform texture spaces, Applied General Topology 4, no. 1 (2003), 157–192. [12] S. Özçağ and Ş. Dost, A categorical view of di-uniform texture spaces, Bol. Soc. Mat. Mexicana 15, no. 3 (2009), 63–80. [13] F. Yıldız, Completeness types for uniformity theory on textures, Filomat 29, no. 1 (2015), 159–178. c© AGT, UPV, 2017 Appl. Gen. Topol. 18, no. 1 217