On Stoltenberg's quasi-uniform completion
aa r X i v : . [ m a t h . GN ] S e p O N S TOLTENBERG ’ S QUASI - UNIFORM C OMPLETION
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Athanasios Andrikopoulos ∗ Dept. of Computer Engineering and InformaticsUniversity of PatrasPatras, 26504, Greece [email protected]
Ioannis Gounaridis
Dept. of Computer Engineering and InformaticsUniversity of PatrasPatras, 26504, Greece [email protected]
Wednesday 2 nd September, 2020 A BSTRACT
In this paper, we give a new completion for quasi-uniform spaces which generalizes the completiontheories of Doitchinov [8] and Stoltenberg [20]. The presented completion theory is very well-behaved and extends the completion theory of uniform spaces in a natural way. That is, the definitionof Cauchy net and the constructed completion coincide with the classical in the case of uniformspaces. The main contribution this completion theory makes is the notion of the cut of nets whichgeneralize the idea of Doitchinov for the notion of D -Cauchy net [1] K eywords Quasi-metric · Quasi-uniformity · Dedekind-MacNeille completion · Cauchy net · Embedding, Complete-ness
The problems of completeness and completion in quasi-uniform spaces were mainly considered in [1], [3], [4], [6], [8],[9], [18], [19], [20]. A satisfactory extension of the completion theory of uniform spaces to arbitrary quasi-uniformspaces first naturally leads to Császár’s double completeness developed in the realm of syntopogenous spaces [6].In this direction, Fletcher and Lindgren have introduced in [9] the notion of bicompleteness, and they prove thatany quasi-uniform space has a bicompletion, called standard bicompletion. It turns out that this concept coincides(for quasi-uniform spaces) with that of double completeness. Since the idea underlying the bicompletion is basicallysymmetric, various authors have tried to construct other, possibly non-symmetric completions as it has naturally arisenfrom the asymmetric character of quasi-uniform spaces. By definition, the notion of completeness of a quasi-uniformspace as well as the construction of the completion depends on the choice of the definition of Cauchy net or Cauchyfilter (often nets and filters lead to equivalent theories). The reason for the difficulty to develop a satisfactory non-symmetric completion theory for the class of all quasi-uniform spaces is due to the struggle to approach the notion ofCauchy net (filter) properly from the case of uniform spaces to the case of quasi-uniform spaces. More precisely, sinceuniform spaces belong to the class of quasi-uniform spaces, according to Doitchinov [8], a notion of Cauchy net inany quasi-uniform space has to be defined in such a manner that this definition provides the properties that convergentnets are Cauchy, and it agrees with the usual definition for uniform spaces. Moreover, the suggested completion mustbe a monotone operator with respect to inclusion and give rise to the usual uniform completion in the uniform case.The problem of defining Cauchy nets or filters in quasi-uniform spaces has been approached by several authors. Theproblem of defining Cauchy nets or filters in quasi-uniform spaces has been initially approached by Császár [6], Sieberand Pervin [19] and Stoltenberg [20]. Császár [6] introduced the notion of Cauchy filter in quasi-uniform spaces andproved that every syntopogenic space can be embedded in complete space. As it concerns Csaszar’s definition ofcauchyness. Isbell [12] noted that the convergent nets were not necessarily Cauchy. Stoltenberg in [20] also gave adefinition of Cauchy net in quasi-uniform spaces which generalize the definition of Kelly [13] for Cauchy sequencesin quasi-metric spaces. According to Stoltenberg’s definition, one can find a Cauchy sequence (net) that is veryinconvenient to regard this sequence as a potentially convergent one by completing the space (see [7, Example 3], [11]. ∗ PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020Sieber and Pervin [19] gave a definition of Cauchy filter in a quasi- uniform space ( X, U ) which admits an equivalentdefinition for nets: A net ( x a ) a ∈ A is Cauchy in ( X, U ) whenever given U ∈ U there is a point x U ∈ X and a ∈ A such that ( x U , x a ) ∈ U for all a ≥ a , a ∈ A . The definition of Sieber-Pervin has been used by many authorsand is usually accepted as the most appropriate way of generalizing the notion of Cauchy net in uniform spaces.According to Doitchinov, the definition of Sieber-Pervin has a serious flaw. Namely, the sequences (nets) dependsnot only on its terms but also on some other points which need not belong to it [7, Example 2]. There are variousgeneralizations to the notion of Cauchy sequence (net) which are based on the definitions of Császár, Stoltenbergand Sieber and Pervin, but, up to now, none of these generalizations can give a satisfying completion theory for allquasi-uniform spaces. Thus, there exist many different notions of quasi-uniform completeness in the literature. Moreprecisely, this problem has been studied in [17], where 7 different notions of “Cauchy sequences (nets)”are presented.By combining the 7 “Cauchy sequences (nets)”with the topologies τ U , and τ U− , we may reach a total of 14 differentdefinitions of “complete space”(considering the symmetry of using the U − instead of U ). Doitchinov [8] developedan interesting completion theory for quiet quasi-metric spaces. What is interesting in Doitchinov’s work is that he hasconsidered a quasi-uniform space as a bitopological space and introduced the concept of conet of a net. Künzi andKivuvu have extended the completion theory of Doitchinov of quasi-pseudometric (quasi-uniform) spaces to arbitraryspaces, denoted B -completion. Andrikopoulos [2] has introduced a new technique, inspired by Dedekind-MacNeillecompletion of rational numbers.In this paper, we give a completion theory based on Stoltenberg’s one which generalizes Doitchinov’s completiontheory and it satisfies all requirements posed by him for a good completion. The technique stands on the constructionof a cut of nets, using Doitchinov’s concept of Cauchy pair of nets. In fact, by Proposition 4.1, to each D -Cauchynet it corresponds a set of pairs of nets-conets which lead to the notion of cut of nets, that is, a pair ( C , D ) where C contains all equivalent nets of the given net and D contains all the conets of the members of C . The notion of U -cutdefined in this paper is a cut of nets ( A , B ) where the members of A contains right U S -Cauchy nets and B containsleft U S -Cauchy nets as they are defined by Stoltenberg. We call the space U -complete if each member of the first classof a U -cut converges and we prove that each quasi-uniform space has a U -completion. The new completion eliminatesthe weaknesses of Stoltenberg’s completion. Let us recall some main notions, basic concepts, definitions and results needed in the paper (see [15], [21]). A quasi-pseudometric space ( X, d ) is a set X together with a non-negative real-valued function d : X × X −→ R (called aquasi-pseudometric) such that, for every x, y, z ∈ X : (i) d ( x, x ) = 0 ; (ii) d ( x, y ) ≤ d ( x, z ) + d ( z, y ) . If d satisfiesthe additional condition (iii) d ( x, y ) = 0 implies x = y , then d is called a quasi-metric on X . A quasi-pseudometric isa pseudometric provided d ( x, y ) = d ( y, x ) . The conjugate of a quasi-pseudometric d on X is the quasi-pseudometric d − given by d − ( x, y ) = d ( y, x ) . By d ∗ we denote the pseudometric given by d ∗ = max { d ( x, y ) , d − ( x, y ) } . Eachquasi-pseudometric d on X induces a topology τ d on X which has as a base the family of d -balls { B d ( x, r ) : x ∈ X, r > } where B d ( x, r ) = { y ∈ X : d ( x, y ) < r } . A quasi-pseudometric space is T if its associated topology τ d is T . In that case axiom (i) and the T -condition can be replaced by (i ′ ) ∀ x, y ∈ X , d ( x, y ) = d ( y, x ) = 0 ⇔ x = y . Inthis case we say that d is a T - quasi-pseudometric . If P is a family of quasi-pseudometrics on the set X , we say that P is a quasi-gauge . The topology τ ( P ) which has as a subbase the family of all balls B ( x, p, ǫ ) with p ∈ P , x ∈ X and ǫ > is called the topology induced on X by the quasi-gauge P (see [16]).A quasi-uniformity on a non-empty set X is a filter U on X × X which satisfies: (i) ∆( X ) = { ( x, x ) | x ∈ X } ⊆ U for each U ∈ U and (ii) given U ∈ U there exists V ∈ U such that V ◦ V ⊆ U . The elements of the filter U are called entourages . If U is a quasi-uniformity on a set X , then U − = { U − | U ∈ U} is also a quasi-uniformity on X calledthe conjugate of U . A uniformity for X is a quasi-uniformity which also satisfies the additional axiom: (iii) For all U ∈ U we have U − ∈ U ( U = U − ). The pair ( X, U ) is called a ( quasi -) uniform space . Given a quasi-uniformity U on X , U ⋆ = U W U − will denote the coarsest uniformity on X which is finer than U . If U ∈ U , the entourage U ∩ U − of U ⋆ will be denoted by U ⋆ . A family B is a base for a quasi-uniformity U if and only if for each U ∈ U there exists B ∈ B such that B ⊆ U . The family B is subbase for U if the family of finite intersections of members of B form a base for U . Every quasi-uniformity U on X generates a topology τ ( U ) . A neighborhood base for each point x ∈ X is given by { U ( x ) | U ∈ U} where U ( x ) = { y ∈ X | ( x, y ) ∈ U } .If ( X, d ) is a quasi-pseudometric space then B = { U d,ǫ | ǫ > } , where U d,ǫ = { ( x, y ) ∈ X × X | d ( x, y ) < ǫ } , is abase for a quasi-uniformity U d for X such that τ d = τ ( U d ) . For each quasi-uniformity U possessing a countable basethere is a quasi-pseudometric d U such that τ ( U ) = τ d U .A function f from a quasi-uniform space ( X, U ) to a quasi-uniform space ( X, V ) is quasi-uniformly continuous if foreach V ∈ V there is U ∈ U such that ( f ( x ) , f ( y )) ∈ V whenever ( x, y ) ∈ U i.e. the set { ( x, y ) | ( f ( x ) , f ( y )) ∈ V} ∈ PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020 U . The function f is a quasi-uniform isomorphism if and only if f is one-to-one onto Y and f − is quasi-uniformlycontinuous. A quasi-uniform space ( X, U ) can be embedded in a quasi-uniform space ( X, V ) when there exists aquasi-uniform isomorphism from ( X, U ) onto a subspace of ( Y, V ) . A sequence ( x n ) n ∈ N in a quasi-pseudometricspace ( X, d ) is called d K - Cauchy (from Kelly [13, Definition 2.10]) if for each ǫ > there is k ∈ N such that d ( x n , x m ) < ǫ for each n ≥ m ≥ k . This is the notion of Cauchy sequence called in [17, Definition 1.iv] a right K- Cauchy sequence . Similarly, a sequence ( x n ) n ∈ N is left K- Cauchy if for each ǫ > there is k ∈ N such that d ( x m , x n ) < ǫ for each n ≥ m ≥ k (see [17, Definition 1.v]). The space ( X, d ) is said to be right (resp. left )K- sequentially complete if each right (resp. left ) K- Cauchy sequence converges in X . According to ([2, Definition 3])a sequence ( x n ) n ∈ N on X is right (resp. left ) d - cofinal to a sequence ( x m ) m ∈ N on X , if for each ε > there exists n ε ∈ N satisfying the following property: for each n ≥ n ε there exists m n ∈ N such that d ( x m , x n ) < ε (resp. d ( x n , x m ) < ε ) whenever m ≥ m n . The sequences ( x n ) n ∈ N and ( x m ) m ∈ N are right (resp. left ) d - cofinal if ( x n ) n ∈ N is right (resp. left) d -cofinal to ( x m ) m ∈ N and vice versa. According to([8, Definition 1]) a sequence ( y m ) m ∈ N is calleda cosequence to ( x n ) n ∈ N , if for any ε > there are n ε , m ε ∈ N such that d ( y m , x n ) < ε when n ≥ n ε , m ≥ m ε .In this case, we write d ( y m , x n ) → or lim m,n d ( y m , x n ) = 0 . Generally speaking, when two sequences ( x n ) n ∈ N and ( y m ) m ∈ N are given in a quasi-pseudometric space ( X, d ) we will write lim n,m d ( y m , x n ) = r if for any ε > there isan N ε such that | d ( y m , x n ) − r | < ε when m, n > N ǫ . We call κ - cut (of sequences) in X ([2, Definition 8]) anordered pair ξ = ( A , B ) of families of right K -Cauchy sequences and left K -Cauchy cosequences, respectively, withthe following propositionerties: (i) For any ( x n ) n ∈ N ∈ A and any ( x m ) m ∈ N ∈ B there holds lim m,n d ( x m , x n ) = 0 ; (ii)Any two members of the family A (resp. B ) are right (resp. left) d -cofinal; (iii) The classes are maximal with respectto set inclusion. We call the member A (resp. B ) first (resp. second ) class of ξ . A κ - Cauchy sequence is a right K -Cauchy sequence which is member of the first class of a κ -cut (see [2, Definition 11]).A directed set is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), withthe additional propositionerty that for any x and y in A there must exist z in A with x ≤ z and y ≤ z . Directedsets are a generalization of nonempty totally ordered sets. That is, all totally ordered sets are directed sets. A net ina topological space X is a function δ : A → X , where A is some directed set. The point δ ( a ) is usually denoted x a and the net is denoted by ( x a ) a ∈ A . A function ϕ : D → A is cofinal in A if for each a ∈ A , there exists some µ ∈ D such that a ≤ ϕ ( µ ) . A subnet of a net δ : A → X is the composition δ ◦ ϕ , where ϕ : D → A is an increasingcofinal function from a directed set D to A . That is: (i) ϕ ( µ ) ≤ ϕ ( µ ) whenever µ ≤ µ ( ϕ is increasing); (ii) ϕ iscofinal in A . For each µ ∈ M , the point δ ◦ ϕ ( µ ) is often written x aµ , and we usually speak of “the subnet ( x aµ ) µ ∈ M of ( x a ) a ∈ A ." A net ( x a ) a ∈ A in quasi-uniform space ( X, U ) is said to be convergent to x ∈ X if for every U ∈ U there exists a U ∈ D such that ( x, x a ) ∈ U for each a ≥ a U . The definition of net generalizes a key result aboutsubsequences: A net ( x a ) a ∈ A converges to x if and only if every subnet of ( x a ) a ∈ A converges to x .Let X be a well-ordered set and let R ( x λ ) , λ ∈ Λ be a propositionosition with domain X . The Principle of TransfiniteInduction asserts that if [ λ<µ R ( x λ ) implies R ( x µ ) , for all µ ∈ Λ , then in fact R ( x λ ) holds for all λ ∈ Λ .Let X a be a set, for each a ∈ A . The Cartesian product of the sets X a is the set Y a ∈ A X a = { x : A → [ a ∈ A X a | x ( a ) ∈ X a , for each a ∈ A } .The value of x ∈ Y a ∈ A at a is usually denoted x a , rather than x ( a ) , and x a is referred to as the ath coordinate of x . Thespace X a in the ath factor space . For each γ ∈ A the map π γ : Y a ∈ A X a → X γ , defined by π γ ( x ) = x γ ,is called the projection map of Y a ∈ A X a on X γ , or the γ th projection map . The Axiom of choice ensure that the Cartesian productof a non-empty collection of non-empty sets is non-empty.Let ( X i , U i ) i ∈ I be a family of quasi-uniform spaces. Let Y i ∈ A X i be the set-theoretic product of the family ( X i ) i ∈ I andlet π j : Y i ∈ A X i −→ X j ( j ∈ I ) be the projection onto ( X j , U j ) . Then the coarsest quasi-uniformity on Y i ∈ A X i thatmakes all projections uniformly continuous is called the product quasi-uniformity. It induces the product topology ofthe topological spaces ( X i , U i ) . 3 PREPRINT - W
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Throughout the paper ( X, U ) will be an arbitrary quasi-uniform space and ( X, d ) will be an arbitrary quasi-pseudometric space, except the cases when it is explicitly stated that the space is T .Stoltenberg [20] has given the following definitions. The index S in our symbolisms is devoted to Stoltenberg. Definition 3.1. ([20, Definition 2.1]). A net ( x a ) a ∈ A in a quasi-unifrom space ( X, U ) is called right U S - Cauchy ( left U S - Cauchy ) if for each U ∈ U there is a U ∈ A such that ( x β , x α ) ∈ U (resp. ( x α , x β ) ∈ U ) whenever α ≥ a U , β ≥ a U , a (cid:3) β and a, β ∈ A . Definition 3.2. ([20, Definition 2.2]). A quasi-uniform space ( X, U ) is U S - complete if and only if each right U S -Cauchy net converges in X with respect to τ U . Definition 3.3.
A net ( x a ) a ∈ A in a quasi-pseudometric space ( X, d ) is called right (resp. left ) d S - Cauchy if for each ǫ > there is a ǫ ∈ A such that d ( x β , x α ) < ǫ (resp. d ( x α , x β ) < ǫ ) whenever α ≥ a ǫ , β ≥ a ǫ , a (cid:3) β and a, β ∈ A . Without loss of generality, we may suppose that for ε ′ ≤ ε , it is a ε ′ ≥ a ε (resp. β ε ′ ≥ β ε ). We say thata quasi-pseudometric space ( X, d ) is d S - complete if and only if every right d S -Cauchy net converges to a point in X with respect to τ d . Similarly, we say that ( X, d ) is U d - complete if and only if every U d -Cauchy net converges to apoint in X with respect to τ U d = τ d .In case of sequences, Definition 3.3 coincides with the definition of the notions of right K -Cauchy sequence, left K -Cauchy sequence, right K -sequentially complete and left K -sequentially complete quasi-pseudometric space, re-spectively. Proposition 3.1. (see [20, Page 229]). Let ( X, d ) be a quasi-pseudometric space. A net ( x a ) a ∈ A in ( X, d ) is a right d S -Cauchy net if and only if ( x a ) a ∈ A is a right ( U d ) S -Cauchy net in ( X, U d ) . Simillarly, the space ( X, d ) is d S -completeif and only if ( X, U d ) is U d -complete. Proof.
By definition, we have that each quasi-pseudometric d on X generates a quasi-uniformity U d with base {{ ( x, y ) ∈ X × X | d ( x, y ) < ǫ }| ǫ > } . Therefore, the implication of the Proposition is an immediate consequenceof the Definitions 3.1-3.3. Definition 3.4. ([8, Definition 1], [7]). Let ( X, U ) (resp. ( X, d ) ) be a quasi-uniform space (quasi-pseudometric space)and let ( x a ) a ∈ A , ( y β ) β ∈ B be two nets in ( X, U ) (resp. ( X, d ) ). The net ( y β ) β ∈ B is called a conet of ( x a ) a ∈ A , if forany U ∈ U (resp. ǫ > ) there are a U ∈ A and β U ∈ B (resp. a ǫ ∈ A and β ǫ ∈ B ) such that ( y β , x a ) ∈ U (resp. d ( y β , x a ) < ǫ ) whenever a ≥ a U , β ≥ β U (resp. a ≥ a ǫ , β ≥ β ǫ ) and a, β ∈ A . Definition 3.5. ([2, Definition 3] ). A net ( x a ) a ∈ A in a quasi-pseudometric space ( X, d ) is called right (resp. left ) d - cofinal to a net ( x β ) β ∈ B on X , if for each ǫ > there exists a ǫ ∈ A satisfying the following property: for each a ≥ a ǫ there exists β a ∈ B such that d ( x β , x a ) < ǫ (resp. d ( x a , x β ) < ǫ ) whenever β ≥ β a . The nets ( x a ) a ∈ A and ( x β ) β ∈ B are right (resp. left ) d - cofinal if ( x a ) a ∈ A is right (resp. left) d -cofinal to ( x β ) β ∈ B and vice versa. Definition 3.6.
Let ( X, d ) be a quasi-pseudometric space. We call δ -cut in X an ordered pair ξ = ( A ξ , B ξ ) of familiesof right d S -Cauchy nets and left d S -Cauchy nets respectively, with the following properties:(i) Any ( x a ) a ∈ A ∈ A ξ has as conet any ( y β ) β ∈ B ∈ B ξ ;(ii) Any two members of the family A ξ (resp. B ξ ) are right (resp. left) d -cofinal.(iii) The classes A ξ and B ξ are maximal with respect to set inclusion. Definition 3.7.
To every x ∈ X we correspond the d -cut φ ( x ) = ( A φ ( x ) , B φ ( x ) ) satisfying the requirements (i)-(iii) ofDefinition 3.6 with the additional condition (iv): The net ( x ) = x, x, x, ... itself, belongs to both of the classes.The notion of δ -cut of nets generalizes the notion of κ -cut of sequences (see [2, Definition 11]). Remark 3.2.
In fact, in the previous definition the members of A φ ( x ) converge to x with respect to τ d and the membersof B φ ( x ) converge to x with respect to τ d − .Let ( X, d ) be a quasi-pseudometric space and let b X denotes the set of all d -cuts in ( X, d ) . Throughtout the paper, forevery ξ ∈ b X , A ξ , B ξ denote the two classes of ξ . In this case, we write ξ = ( A ξ , B ξ ) . Remark 3.3.
If the space ( X, d ) is T , then the function φ defined above is an injective function (one-to-one) of X into b X . Indeed, let x, y ∈ X be such that φ ( x ) = φ ( y ) . Then, ( x ) , ( y ) ∈ A φ ( x ) ∩ B φ ( x ) ∩ A φ ( y ) ∩ A φ ( y ) . Thus, d ∗ ( x, y ) = 0 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020which implies that x = y . Generally, each quasi-pseudometric space ( X, d ) defines a T quasi-pseudometric space ( X ∗ , d ∗ ) . Indeed, let R be the equivalence relation in X defined by: xRy if and only if d ( x, y ) = 0 = d ( y, x ) . Letalso X ∗ = X/R = { [ x ] | x ∈ X } = {{ z ∈ X | zRx } | x ∈ X } be the quotient space (the set of equivalence classes).It is convenient to introduce the function π : X → X ∗ defined by π ( x ) = { [ y ] | x ∈ [ y ] } . Since each x ∈ X iscontained in exactly one equivalence class we have π ( x ) = [ x ] , that is, the function x → [ x ] is well defined. In orderto ensure that the projection map be continuous we put an obligation on the topology we assign to X ∗ : If a set G isopen in X ∗ then π − ( G ) is open in X . We define the quotient topology on X ∗ by letting all sets G that pass this testbe admitted. On other words, a set G is open in X ∗ if and only if π − ( G ) is open in X . The quotient topology on X ∗ is the finest topology on X ∗ for which the projection map π is continuous. Let d ∗ : X ∗ × X ∗ → R be a functiondefined by d ∗ ([ x ] , [ y ]) = d ( x, y ) . Then, it is easy to check that d ∗ is a T quasi-pseudometric in X ∗ that yields thequotient topology in X ∗ . Definition 3.8.
Let ( X, d ) be a quasi-pseudometric space. We call δ - Cauchy net any right d S -Cauchy net member ofthe first class of a δ -cut. The space ( X, d ) is δ - complete if and only if each δ -Cauchy net converges in X . Definition 3.9.
A net ( x a ) a ∈ A in a quasi-uniform space ( X, U ) is called right (resp. left ) U - cofinal to a net ( x β ) β ∈ B on X , if for each U ∈ U there exists a U ∈ A satisfying the following property: for each a ≥ a U there exists β a ∈ B such that ( x β , x a ) ∈ U (resp. ( x a , x β ) ∈ U ) whenever β ≥ β a . The nets ( x a ) a ∈ A and ( x β ) β ∈ B are right (resp. left ) U - cofinal if ( x a ) a ∈ A is right (resp. left) U -cofinal to ( x β ) β ∈ B and vice versa. Definition 3.10.
Let ( X, U ) be a quasi-uniform space. We call U -cut in X an ordered pair ξ = ( A ξ , B ξ ) of familiesof right U S -Cauchy nets and left U S -Cauchy nets respectively, with the following properties:(i) Any ( x a ) a ∈ A ∈ A ξ has as conet any ( y β ) β ∈ B ∈ B ξ ;(ii) Any two members of the family A ξ (resp. B ξ ) are right (resp. left) U -cofinal.(iii) The classes A ξ and B ξ are maximal with respect to set inclusion. Definition 3.11.
To every x ∈ X we correspond the U -cut φ ( x ) = ( A φ ( x ) , B φ ( x ) ) satisfying the requirements (i)-(iii)of Definition 3.6 with the additional condition (iv): The members of A φ ( x ) converge to x with respect to τ U and themembers of B φ ( x ) converge to x with respect to τ U− .According to Definition 3.11, the net ( x ) = x, x, x, ... itself, belongs to both of the classes A φ ( x ) and B φ ( x ) . Definition 3.12.
Let ( X, U ) be a quasi-uniform space. We call U - Cauchy net any right U S -Cauchy net member of thefirst class of a U -cut. The space ( X, U ) is U - complete if and only if each U -Cauchy net converges in X .Two δ -Cauchy (resp. U -Cauchy) nets ( x a ) a ∈ A and ( x β ) β ∈ B in a quasi-pseudometric space ( X, d ) (resp. quasi-uniformspace ( X, U ) ) are called δ - equivalent (resp. U - equivalent ) if every conet to ( x a ) a ∈ A is a conet to ( x β ) β ∈ B and viceversa.It is easy to see that δ -equivalence (resp. U -equivalence) defines an equivalence relation on ( X, d ) (resp. ( X, U ) ).Hence, A is the equivalence class of the δ -Cauchy nets (resp. U -Cauchy nets) that are considered to be equivalent bythis equivalence relation.In metric (resp. uniform) spaces the classes A and B coincide with a well known equivalent classes of Cauchy nets. Proposition 3.4.
Let ( x a ) a ∈ A be a right (resp. left) d S -Cauchy net in a quasi-pseudometric space ( X, d ) with a subnet ( x aγ ) γ ∈ Γ . Then, ( x a ) a ∈ A and ( x aγ ) γ ∈ Γ are right (resp. left) d -cofinal. Proof.
Let ( x a ) a ∈ A be a right d S -Cauchy net in ( X, d ) and ( x aγ ) γ ∈ Γ be a subnet of it. Let ǫ > be given. Then, thereexists a ǫ ∈ A such that for each a, a ′ ∈ A with a ≥ a ǫ , a ′ ≥ a ǫ and a ′ (cid:3) a , we have d ( x a , x a ′ ) < ǫ . Let a γ > a ǫ for some γ ∈ Γ and a aγ = a γ . Then, for each a > a γ ( a γ (cid:3) a ) we have d ( x a , x aγ ) < ǫ . Hence, ( x aγ ) γ ∈ Γ is right d -cofinal to ( x a ) a ∈ A . On the other hand, let a > a ǫ for some a ∈ A . Let also ( a γ ) a = a γ for some a γ > a , γ ∈ Γ .Then, for each γ ′ > γ we have d ( x aγ ′ , x a ) < ǫ which implies that ( x a ) a ∈ A is right d -cofinal to ( x aγ ) γ ∈ Γ . The case ofthe left d S -Cauchy net is simillar. Proposition 3.5.
In every quasi-pseudometric space ( X, d ) two right d -cofinal nets have the same conets. Proof.
Let ( x a ) a ∈ A , ( x β ) β ∈ B be two right d -cofinal nets. Suppose that ( y σ ) σ ∈ Σ is a conet of ( x a ) a ∈ A . Fix ǫ > . Thenthere exist σ ǫ ∈ Σ and a ǫ ∈ A such that d ( y σ , x a ) < ǫ for σ ≥ σ ǫ , a ≥ a ǫ . On the other hand, there is β ǫ ∈ B PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020with the following property: for each β ≥ β ǫ there exists a β ∈ A such that d ( x a , x β ) < ǫ whenever a ≥ a β . If a ∗ = max { a ǫ , a β } , then from d ( y σ , x a ∗ ) < ǫ and d ( x a ∗ , x β ) < ǫ we conclude that d ( y σ , x β ) < ǫ ǫ ǫ for σ ≥ σ ǫ and β ≥ β ǫ .Similarly we can prove the following proposition. Proposition 3.6.
In every quasi-pseudometric space ( X, d ) , two left d -cofinal nets are conets of the same nets.The following two corollaries is an immediate consequence of Propositions 3.5 and 3.6. Corollary 3.7.
In every quasi-pseudometric space ( X, d ) , two right (left) d -cofinal nets have the same limit points for τ ( d ) (resp. τ ( d − ) ). Corollary 3.8.
Let ( X, d ) be a quasi-pseudometric space and let ξ, ξ ′ ∈ b X . Then, A ξ ∩ A ξ ′ = ∅ implies A ξ = A ξ ′ .Let ( X, d ) be a quasi-pseudometric space and let b X be the set of all d -cuts in ( X, d ) . Throughtout the paper, for every ξ ∈ b X , A ξ , B ξ denote the two classes of ξ . In this case, we write ξ = ( A ξ , B ξ ) . Definition 3.13.
Let ( X, d ) be a quasi-pseudometric space. Suppose that r is a nonnegative real number, ξ ′ , ξ ′′ ∈ b X , ( x a ) a ∈ A ∈ A ξ ′ and ( x γ ) γ ∈ Γ ∈ B ξ ′′ . We put b d ( ξ ′ , ξ ′′ ) ≤ r if:(i) A ξ ′ = A ξ ′′ or(ii) For each ǫ > there are a ǫ ∈ A , γ ε ∈ Γ such that d ( x a , x γ ) < r + ǫ (1)when a ≥ a ε and γ ≥ γ ε . If ξ ′ = φ ( x ) for some x ∈ X , then the net ( x a ) a ∈ A always coincides with the fixed net, forwhich x a = x for all a ∈ A . Then, we let b d ( ξ ′ , ξ ′′ ) = inf { r | b d ( ξ ′ , ξ ′′ ) ≤ r } . (2) Proposition 3.9.
The truth of b d ( ξ ′ , ξ ′′ ) ≤ r in Definition 3.13(ii) depends only on ξ ′ , ξ ′′ , and r ; it does not depend onthe choice of the nets ( x a ) a ∈ A and ( x γ ) γ ∈ Γ . Proof.
Let ( x a ) a ∈ A and ( x β ) β ∈ B be two right U S -Cauchy nets of the class A ξ ′ and ( x γ ) γ ∈ Γ and ( x δ ) δ ∈ ∆ be two right U S -Cauchy nets of the class A ξ ′′ . Then, for each ε > there are a ε ∈ A and γ ε ∈ Γ such that d ( x a , x γ ) < r + ε when a ≥ a ε and γ ≥ γ ε . Choose an arbitrary positive number ε ′ so that < ε ′ < ε . Then, there are a ε ′ ∈ A and γ ε ′ ∈ Γ such that d ( x a , x γ ) < r + ε ′ when a ≥ a ε ′ and γ ≥ γ ε ′ . Since ( x a ) a ∈ A is left d -cofinal to ( x β ) β ∈ B , there is a ′ ε ∈ A satisfying the followingproperty: For each a ≥ a ′ ε there exists β a ∈ B such that d ( x β , x a ) < ε whenever β ≥ β a . Fix an a ′ ε ≥ a ε ′ and let a ′ ε = a ∗ . Then, we have d ( x β , x γ ) ≤ d ( x β , x ∗ ) + d ( x ∗ , x γ ) < r + ε ′ + ε whenever β ≥ β a ′ ε and γ ≥ γ ε ′ .Similarly, since ( x δ ) δ ∈ ∆ is left d -cofinal to ( x γ ) γ ∈ Γ , there is δ ε ∈ ∆ satisfying the following property: For each δ ≥ δ ε there exists γ δ ∈ Γ such that 6 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020 d ( x γ , x δ ) < ε whenever γ ≥ γ δ . Let γ ∗ = max { γ ε ′ , γ δ } . Then, d ( x β , x δ ) ≤ d ( x β , x γ ∗ ) + d ( x γ ∗ , x δ ) < r + ε ′ + ε ε < r + ε whenever β ≥ β a ′ ε and δ ≥ δ ε . Proposition 3.10.
Let ξ ′ , ξ ′′ ∈ b X , ( x a ) a ∈ A ∈ A ξ ′ and ( x γ ) γ ∈ Γ ∈ A ξ ′′ and A ξ ′ = A ξ ′′ . Then, b d ( ξ ′ , ξ ′′ ) = lim a,γ d ( x a , x γ ) . Proof.
Let b d ( ξ ′ , ξ ′′ ) = r . Then, for any ε > there are a ε ∈ A and γ ε ∈ Γ such that d ( x a , x γ ) < r + ε whenever a ≥ a ε and γ ≥ γ ε . To prove that r − ε < d ( x a , x γ ) for a ≥ a ε and γ ≥ γ ε , suppose to the contrarythere exist a subnet ( x aλ ) λ ∈ Λ of ( x a ) a ∈ A and a subnet ( x γµ ) µ ∈ M of ( x γ ) γ ∈ Γ such that for all x aλ , x γµ there holds d ( x aλ , x γµ ) ≤ r − ε . Then, by Propositions 3.4, 3.5, 3.9 and Definition 3.13 we have that b d ( ξ ′ , ξ ′′ ) ≤ r − ε , acontradiction. Therefore, we have b d ( ξ ′ , ξ ′′ ) = r = lim a,γ d ( x a , x γ ) . Proposition 3.11. b X is quasi-pseudometric. Proof.
From Definition 3.13 it follows immediately that b d ( ξ, ξ ) = 0 and b d ( ξ, ξ ′ ) ≥ for all ξ, ξ ′ ∈ b X . To prove that b d satisfies the triangle inequality, let ξ, ξ ′ , ξ ′′ ∈ b X . We have four cases to consider:(i) A ξ = A ξ ′ and A ξ ′ = A ξ ′′ . Suppose that b d ( ξ, ξ ′ ) = r , b d ( ξ ′ , ξ ′′ ) = r , ( x a ) a ∈ A ∈ A ξ , ( x β ) β ∈ B ∈ A ξ ′ and ( x γ ) γ ∈ Γ ∈ A ξ ′′ . Then, for any ε > there are a ε ∈ A and β ε ∈ B such that d ( x a , x β ) < r + ε whenever a ≥ a ε and β ≥ β ε . Similarly, there are β ′ ε ∈ B and γ ε ∈ Γ such that d ( x β , x γ ) < r + ε whenever β ≥ β ′ ε and γ ≥ γ ε . Let B = max { β ε , β ′ ε } . Then, d ( x a , x γ ) ≤ d ( x a , x βM ) + d ( x βM , x γ ) < r + r + ε . Hence, according to Definition3.13, we have b d ( ξ, ξ ′′ ) ≤ r + r = b d ( ξ, ξ ′ ) + b d ( ξ ′ , ξ ′′ ) .(ii) A ξ = A ξ ′ and A ξ ′ = A ξ ′′ . Suppose that b d ( ξ, ξ ′ ) = r . Since A ξ ′ = A ξ ′′ , Definition 3.13 implies that b d ( ξ, ξ ′ ) ≤ r and b d ( ξ ′ , ξ ′′ ) = 0 . Therefore, b d ( ξ, ξ ′′ ) ≤ r = b d ( ξ, ξ ′ ) + b d ( ξ ′ , ξ ′′ ) .(iii) A ξ = A ξ ′ and A ξ ′ = A ξ ′′ .(iv) A ξ = A ξ ′ and A ξ ′ = A ξ ′′ . This case is trivial. Proposition 3.12.
Let ( X, d ) be a quasi-pseudometric space. Then, for any x, y ∈ X we have b d ( φ ( x ) , φ ( y )) = d ( x, y ) . Proof.
We have that ( x ) ∈ A φ ( x ) and ( y ) ∈ A φ ( y ) . If A φ ( x ) = A φ ( y ) , then b d ( φ ( x ) , φ ( y )) = 0 = d ( x, y ) . Otherwise,Proposition 3.10 implies that b d ( φ ( x ) , φ ( y )) = d ( x, y ) . 7 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020
Proposition 3.13.
For any ξ = ( A ξ , B ξ ) ∈ b X , ( x a ) a ∈ A ∈ A ξ implies that b d ( ξ, φ ( x a )) → and ( x β ) β ∈ B ∈ B ξ implies that b d ( φ ( x β ) , ξ ) → . Proof.
We now prove that if ( x a ) a ∈ A ∈ A ξ then φ ( x a ) converges to ξ . Fix an ε > . Since ( x a ) a ∈ A is a right d S -Cauchy net, there exists a ε ∈ A such that d ( x a , x a ′ ) < ε , whenever a ≥ a ε , a ′ ≥ a ε and a ′ (cid:3) a. (3)Let ( x σ ) σ ∈ Σ ∈ A ξ . Then, since ( x a ) a ∈ A and ( x σ ) σ ∈ Σ right d -cofinal we have the property: There exists b a ε ∈ A and σ b aε ∈ Σ , such that d ( x σ , x a ) < ǫ , whenever a ≥ b a ε and σ ≥ σ b aε . (4)Let a ′′ ∈ A be such that a ′′ ≥ a ε and a ′′ ≥ b a ε ( A is directed). Fix an a ≥ a ε and a right d S -Cauchy net ( x ρ ) ρ ∈ P ∈A φ ( xa ) . There exists ρ ε ∈ P such that d ( x a , x ρ ) < ǫ , whenever ρ ≥ ρ ε . (5)We have two cases to consider - the case where a ≥ a ′′ and the case where a (cid:3) a ′′ .(i) Let a ≥ a ′′ . Then, since a ≥ a ′′ ≥ a ε and a ≥ a ′′ ≥ b a ε we have d ( x σ , x ρ ) ≤ d ( x σ , x a ) + d ( x a , x ρ ) < ǫ ǫ < ǫ < ǫ, whenever σ ≥ σ b aε and ρ ≥ ρ ǫ . (6)(ii) Let a (cid:3) a ′′ . Then, since a ′′ ≥ b a ε and a, a ′′ ≥ a ε with a (cid:3) a ′′ we have that d ( x σ , x ρ ) ≤ d ( x σ , x a ′′ ) + d ( x a ′′ , x a ) + d ( x a , x ρ ) < ǫ ǫ ǫ ǫ, whenever σ ≥ σ b aε and ρ ≥ ρ ǫ . (7)Since ( x σ ) σ ∈ Σ ∈ A ξ , ( x ρ ) ρ ∈ P ∈ A φ ( xa ) , then by (6) and (7) and Definition 3.13, we conclude that d ( ξ, φ ( x a )) < ε for a ≥ a ε which implies that b d ( ξ, φ ( x a )) → . (8)Similarly we can prove that b d ( φ ( x β ) , ξ ) → . Proposition 3.14.
Let ( X, d ) be a quasi-pseudometric space and let ( ξ a ) a ∈ A be a non-constant right b d S -Cauchy netin ( b X, b d ) without last element. Let also a ∗ ∈ A be such that for each a ≥ a ∗ , a ′ ≥ a ∗ and a ′ (cid:3) a there holds b d ( ξ a , ξ a ′ ) = 0 . Then, there exists a right d S -Cauchy net ( t σ ) σ ∈ Σ in ( X, d ) such that ( ξ a ) a ∈ A and ( φ ( t σ )) σ ∈ Σ are right b d -cofinal nets. Proof. ( α ) The Construction of the right d S -Cauchy net ( t σ ) σ ∈ Σ in ( X, d ) . Let ( ξ a ) a ∈ A and a ∗ be as in the proposition.Since b d ( ξ a , ξ a ′ ) = 0 , for each ǫ > we have that b d ( ξ a , ξ a ′ ) < ǫ whenever a ≥ a ∗ , a ′ ≥ a ∗ and a ′ (cid:3) a. (9)Let (Γ , ≤ ) be cofinal well-ordered subset of ( A, ≤ ) (the existence of such Γ follows from the axiom of choice).Consider the subnet ( ξ aγ ) γ ∈ Γ = ( ξ γ ) γ ∈ Γ of ( ξ a ) a ∈ A . Then, ( ξ γ ) γ ∈ Γ is cofinal to ( ξ a ) a ∈ A and by (9) we have that b d ( ξ aγ , ξ aγ ′ ) < ǫ whenever γ ≥ γ ′ ≥ a ∗ . (10)For any γ ∈ Γ , let ξ γ = ( A ξγ , B ξγ ) and ( x ρ ( kγ ) ) ρ ( kγ ) ∈ Pkγ be a member of A ξγ where k γ denote the different right d S -Cauchy nets of A ξγ and ρ ( k γ ) denote the indices sets of k γ . Let also ρ ǫ ( k γ ) be the smallest index with the property d ( x ρ ( kγ ) , x ρ ′ ( kγ ) ) < ǫ ρ ( k γ ) ≥ ρ ǫ ( k γ ) , ρ ′ ( k γ ) ≥ ρ ǫ ( k γ ) and ρ ′ ( k γ ) (cid:3) ρ ( k γ ) . (11)For each γ ∈ Γ fix an k ∗ γ ∈ P k ∗ γ . Without loss of generality (see Remark 3.2), we can assume that: For each ǫ ′ ≤ ǫ we have that ρ ǫ ( k ∗ γ ) ≤ ρ ǫ ′ ( k ∗ γ ) . (12)8 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020Let γ > γ ′ and let ǫ > . Then, b d ( ξ γ , ξ γ ′ ) = 0 implies that there exists ρ ǫ ( k ∗ γ ) ∈ P k ∗ γ , ρ ǫ ′ ( k ∗ γ ′ ) ∈ P k ∗ γ ′ such that d ( x ρ ( k ∗ γ ) , x ρ ( k ∗ γ ′ ) ) < ǫ ρ ( k ∗ γ ) ≥ ρ ǫ ( k ∗ γ ) and ρ ( k ∗ γ ′ ) ≥ ρ ǫ ′ ( k ∗ γ ′ ) . (13)We advance to the construction of the demanded right d S -Cauchy net ( t λ ) λ ∈ Λ in ( X, d ) by using transfinite inductionon the well-ordered set (Γ , ≤ ) . Let γ ≥ γ ≥ a ∗ for some γ , γ ∈ Γ and let ǫ > . Then, d ( x ρ ( k ∗ γ , x ρ ( k ∗ γ ) < ǫ ρ ( k ∗ γ ) ≥ ρ ǫ ( k ∗ γ ) and ρ ( k ∗ γ ) ≥ ρ ǫ ( k ∗ γ ) . (14)Let e ρ ǫ ( k ∗ γ ) = ρ ǫ ( k ∗ γ ) and e ρ ǫ ( k ∗ γ ) = min { ρ ǫ ( k ∗ γ ) , ρ ǫ ( k ∗ γ ) } . Then, d ( x ρ ( k ∗ γ , x ρ ( k ∗ γ ) < ǫ < ǫ whenever ρ ( k ∗ γ ) ≥ e ρ ǫ ( k ∗ γ ) and ρ ( k ∗ γ ) ≥ e ρ ǫ ( k ∗ γ ) . (15)We call equation (15), T (1) -Property.Let γ ∈ Γ such that γ > γ > γ . Then, if e ρ ǫ ( k ∗ γ ) = min { ρ ǫ ( k ∗ γ ) , ρ ǫ ( k ∗ γ ) } , as in (15), it is easy to check that d ( x ρ ( k ∗ γ , x ρ ( k ∗ γ ) < ǫ and d ( x ρ ( k ∗ γ , x ρ ( k ∗ γ ) < ǫ whenever ρ ( k ∗ γ ) ≥ e ρ ǫ ( k ∗ γ ) , ρ ( k ∗ γ ) ≥ e ρ ǫ ( k ∗ γ ) and ρ ( k ∗ γ ) ≥ e ρ ǫ ( k ∗ γ ) . (16)We call equation (16), T (2) -Property.Let γ ∈ Γ be an ordinal which has the T ( γ ) -Property, that is: For each γ ′ < γ we have d ( x ρ ( k ∗ γ ) , x ρ ( k ∗ γ ′ ) ) < ǫ whenever ρ ( k ∗ γ ) ≥ e ρ ǫ ( k ∗ γ ) and ρ ( k ∗ γ ′ ) ≥ e ρ ǫ ( k ∗ γ ′ ) . (17)According to transfinite induction, if T ( γ ) is true whenever T ( γ ′ ) is true for all γ ′ < γ , then T ( γ ) is true for all γ .We follow two steps:(i) Successor case:
Prove that for any successor ordinal γ + 1 , T ( γ + 1) follows from T ( γ ) .(ii) Limit case:
Prove that for any limit ordinal γ , T ( γ ) follows from [ T ( γ ′ ) for all γ ′ < γ ] . Step (i) . Let γ be a successor ordinal. Let also δ be an ordinal such that γ < δ . If δ is a limit ordinal, then there existsan ordinal ζ ′ such that γ < ζ ′ < δ , otherwise, there exists an ordinal ζ such that γ < δ < ζ . We only prove that T ( γ + 1) follows from T ( γ ) where γ + 1 = ζ (the case γ < ζ ′ < δ is similar for γ + 1 = δ ).By repeating the steps (13)-(16) above for γ ′ , δ, ζ where γ ′ ≤ γ , instead of γ , γ , γ , considering them with samelayout, we conclude that d ( x ρ ( k ∗ ζ ) , x ρ ( k ∗ γ ′ ) ) < ǫ whenever ρ ( k ∗ ζ ) ≥ e ρ ǫ ( k ∗ ζ ) and ρ ( k ∗ γ ) ≥ e ρ ǫ ( k ∗ γ ′ ) . (18)Therefore, T ( γ + 1) = T ( ζ ) holds. Step (ii) . Let γ be a limit ordinal. As it is well-known, an ordinal γ is a limit ordinal if and only if there is an ordinal lessthan γ , and whenever γ ′ is an ordinal less than γ , then there exists an ordinal γ ′′ such that γ ′ < γ ′′ < γ . Therefore, byrepeating the steps (13)-(16) above for γ ′ , γ ′′ , γ instead of γ , γ , γ , considering them with same layout, we concludethat d ( x ρ ( k ∗ γ ) , x ρ ( k ∗ γ ′ ) ) < ǫ whenever ρ ( k ∗ γ ) ≥ e ρ ǫ ( k ∗ γ ) and ρ ( k ∗ γ ′ ) ≥ e ρ ǫ ( k ∗ γ ′ ) . (19)Therefore, T ( γ ) holds. It follows that T ( γ ) is true for all γ ∈ b Γ , where b Γ is cofinal to Γ .Let e A = { ( γ, ǫ ) | γ ∈ b Γ , ǫ > } . We define an order on e A as follows: ( γ ′ , ǫ ′ ) ≤ ( γ, ǫ ) if and only if (i) γ ′ < γ or (ii) γ ′ = γ and ǫ < ǫ ′ . (20)Clearly, e A is directed with respect to ≤ . Define the net ( x e ρǫ ( k ∗ γ ) ) ( γ,ǫ ) ∈ e A . By (12), for each ǫ ′ < ǫ , we have e ρ ǫ ( k ∗ γ ) ≤ e ρ ǫ ′ ( k ∗ γ ) . Therefore, by (12) and T ( γ ) property we conclude that ( x e ρǫ ( k ∗ γ ) ) ( γ,ǫ ) ∈ e A is a right d S -Cauchy net. We now9 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020prove that the right b d S -Cauchy nets ( ξ a ) a ∈ A and ( φ ( x e ρǫ ( k ∗ γ ) )) ( γ,ǫ ) ∈ e A are left b d -cofinal. Since ( ξ a ) a ∈ A and ( ξ γ ) γ ∈ Γ areleft b d -cofinal and ( ξ γ ) γ ∈ Γ and ( ξ γ ) γ ∈ b Γ are left b d -cofinal, to prove that ( ξ a ) a ∈ A and ( φ ( x e ρǫ ( k ∗ γ ) )) ( γ,ǫ ) ∈ e A are left b d -cofinal,we have to prove that ( ξ γ ) γ ∈ b Γ and ( φ ( x e ρǫ ( k ∗ γ ) )) ( γ,ǫ ) ∈ e A are right b d -cofinal.In order to prove that ( ξ γ ) γ ∈ b Γ and ( φ ( x e ρǫ ( k ∗ γ ) )) ( γ,ǫ ) ∈ e A are right b d S -cofinal, let ǫ ∗ > and γ ǫ ∗ > a ∗ . By construction,we have d ( x e ρǫ ( k ∗ γ ) , x ρ ( k ∗ γǫ ∗ ) ) < ǫ , ( γ, ǫ ) ∈ e A, γ > γ ǫ ∗ , ǫ > ρ ( k ∗ γ ǫ ∗ ) ≥ e ρ ǫ ( k ∗ γ ǫ ∗ ) . (21)Since each ( x ρ ( kγǫ ∗ ) ) ρ ( kγǫ ∗ ) ∈ Pkγǫ ∗ ∈ A ξγǫ ∗ is right d S -cofinal to ( x ρ ( k ∗ γǫ ∗ ) ) ρ ( k ∗ γǫ ∗ ) ∈ Pk ∗ γǫ ∗ ∈ A ξγǫ ∗ we have that d ( x ρ ( k ∗ γǫ ∗ ) , x ρ ( kγǫ ∗ ) ) < ǫ , whenever ρ ( k ∗ γ ǫ ∗ ) ≥ ρ ( k ∗ γ ǫ ∗ ) and ρ ( k γ ǫ ∗ ) ≥ ρ ( k γ ǫ ∗ ) . (22)It follows that d ( x e ρǫ ( k ∗ γ ) , x ρ ( kγǫ ∗ ) ) < ǫ < ǫ, whenever ρ ( k ∗ γ ǫ ∗ ) ≥ ρ ( k ∗ γ ǫ ∗ ) and ρ ( k γ ǫ ∗ ) ≥ ρ ( k γ ǫ ∗ ) . (23)which implies that d ( φ ( x e ρǫ ( k ∗ γ ) ) , ξ γǫ ∗ ) < ǫ, whenever ( γ, ǫ ) > ( γ ǫ ∗ , ǫ ∗ ) and ρ ( k γ ǫ ∗ ) ≥ ρ ( k γ ǫ ∗ ) . (24)Therefore ( φ ( x e ρǫ ( k ∗ γ ) )) ( γ,ǫ ) ∈ e A is right b d S -cofinal to ( ξ γ ) γ ∈ b Γ . On the other hand, in view of Definition ?? and bythe construction of x e ρǫ ( k ∗ γ ) immediately follows that ( ξ γ ) γ ∈ b Γ right b d S -cofinal to ( φ ( x e ρǫ ( k ∗ γ ) )) ( γ,ǫ ) ∈ e A . Therefore, therequired net ( t σ ) σ ∈ Σ of the hypothesis is the net ( x e ρǫ ( k ∗ γ ) ) ( γ,ǫ ) ∈ e A .If the cardinality of index set A in Proposition 3.14 equals to the cardinality ℵ of the set of all natural numbers, thenwe have the following corollary (see also [2, Proposition 25]). Corollary 3.15.
Let ( X, d ) be a quasi-pseudometric space and let ( ξ n ) n ∈ N be a non-constant right K -Cauchy sequencein ( b X, b d ) without last element. Then, there exists a right K -Cauchy sequence ( t ν ) ν ∈ N in ( X, d ) such that the sequences ( ξ n ) n ∈ N and ( φ ( t ν )) ν ∈ N are right b d -cofinal sequences. Proof.
Let let ( ξ n ) n ∈ N be a non-constant right K -Cauchy sequence in ( b X, b d ) without last element and let ( x e ρǫ ( k ∗ γ ) ) ( γ,ǫ ) ∈ e A be as in Proposition 3.14. Since ( ξ n ) n ∈ N is a sequence we have that Γ ⊆ N and e A ⊆ N . In thiscase we put ( x e ρǫ ( k ∗ γ ) ) ( γ,ǫ ) ∈ e A = ( x e ρǫ ( k ∗ γ ( n ) ) ) ( γ ( n ) ,ǫ ) ∈ N . We define an order on N as follows: ( γ ( n ′ ) , ǫ ′ ) ≤ ( γ ( n ) , ǫ ) if and only if (i) n ′ < n or (ii) n ′ = n and ǫ < ǫ ′ . (25)Clearly, e N = { ( γ ( n ) , ǫ ) | γ ( n ) ∈ N , ǫ > } ⊆ N is a linearly ordered set with respect to ≤ . Define the sequence ( x e ρǫ ( k ∗ γ ( n ) ) ) ( γ ( n ) ,ǫ ) ∈ N . By Proposition 3.14 we have that ( ξ n ) n ∈ N and ( φ ( x e ρǫ ( k ∗ γ ( n )) )) ( γ ( n ) ,ǫ ) ∈ N are right b d -cofinal se-quences. Proposition 3.16.
Let ( X, d ) be a quasi-pseudometric space and let ( ξ a ) a ∈ A be a non-constant right b d S -Cauchy netin ( b X, b d ) without last element. Then, there exists a right d S -Cauchy net ( t σ ) σ ∈ Σ in ( X, d ) such that the nets ( ξ a ) a ∈ A and ( φ ( t σ )) σ ∈ Σ are right b d -cofinal nets. Proof.
Let ( b X, b d ) and ( ξ a ) a ∈ A be as in the assumptions of the Proposition. Without loss of generality we can assumethat a = a ′ implies that A ξa = A ξa ′ . Pick a ∈ A such that b d ( ξ a , ξ a ′ ) ≤ whenever a ≥ a , a ′ ≥ a and a ′ (cid:3) a .Given a n ∈ A such that b d ( ξ a , ξ a ′ ) ≤ n whenever a ≥ a ′ ≥ a n , choose a n +1 ∈ A such that a n +1 ≥ a n and b d ( ξ a , ξ a ′ ) ≤ n +1 whenever a ≥ a ′ ≥ a n +1 . Then, the sequence ( ξ an ) n ∈ N is a right K -Cauchy sequence in ( b X, b d ) .We have two cases to consider; (i) There exists a ∗ ∈ A such that a ∗ > a n for each n ∈ N ; (ii) For each a ∈ A there exists n ∈ N and a n ∈ N such that a >/ a n . In case (i), for each a, a ′ ≥ a ∗ ≥ a n , n ∈ N and a ′ (cid:3) a , we10 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020have b d ( ξ a , ξ a ′ ) ≤ n for all n ∈ N . Hence, b d ( ξ a , ξ a ′ ) = 0 . Thus, Proposition 3.14 ensures the existence of a right d S -Cauchy net ( t σ ) σ ∈ Σ in ( X, d ) such that the nets ( ξ a ) a ∈ A and ( φ ( t σ )) σ ∈ Σ are right b d -cofinal nets. In case (ii), since ( ξ an ) n ∈ N is a subsequence of ( ξ a ) a ∈ A Proposition 3.4 implies that ( ξ an ) n ∈ N and ( ξ a ) a ∈ A are right b d -cofinal. On theother hand, Corollary 3.15 implies that there is right K -Cauchy sequence ( t ν ) ν ∈ N in ( X, d ) such that the sequences ( ξ an ) n ∈ N and ( φ ( t ν )) ν ∈ N are right b d -cofinal nets. It follows that ( ξ a ) a ∈ A and ( φ ( t ν )) ν ∈ N are right b d -cofinal nets. Corollary 3.17.
A quasi-metric space ( X, d ) is δ -complete if and only if every right K -Cauchy sequence convergesto a point of ( X, d ) . Proof.
If a quasi-metric space is δ -complete, then each δ -Cauchy net converges in X . Therefore, each δ -Cauchysequence converges in X .Conversely, suppose that ( X, d ) is a quasi-metric space in which every δ -Cauchy sequence converges to a point of X .Let ( x a ) a ∈ A be a δ -Cauchy net in X . Then, by Proposition 3.16 we have two cases to consider: ( a ) d ( x a , x a ′ ) = 0 ,where a, a ′ ≥ a for some a ∈ A and a ′ (cid:3) a ; ( b ) There exists a subsequence ( x a n ) n ∈ N of ( x a ) a ∈ A such that ( x a ) a ∈ A and ( x a n ) n ∈ N are right d -cofinal. In case ( a ) we have x a = x a for all a ∈ A with a (cid:3) a . Therefore, ( x a ) a ∈ A converges to x a . In case ( b ), ( x a n ) n ∈ N converges to a point l ∈ X . By Corollary 3.7 we have that ( x a ) a ∈ A converges to l as well.By using the dual version of Proposition 3.14, Corollary 3.15 and Proposition 3.16 for left d S -Cauchy nets we havethe following proposition. Proposition 3.18.
Let ( X, d ) be a quasi-pseudometric space and let ( η β ) β ∈ B be a non-constant left b d S -Cauchy net in ( b X, b d ) without last element. Then, there exists a left d S -Cauchy net ( t ρ ) ρ ∈ P in ( X, d ) such that the nets ( η β ) β ∈ B and ( φ ( t ρ )) ρ ∈ P are left b d -cofinal nets. Theorem 3.19.
Every quasi-pseudometric space ( X, d ) has a δ -completion. Proof.
Let ( ξ a ) a ∈ A be a δ -Cauchy net in the space ( b X, b d ) . Then, by definition 3.8, there exists a δ -cut b ξ ∈ b X suchthat ( ξ a ) a ∈ A ∈ A b ξ . Let b ξ = ( A b ξ , B b ξ ) where A b ξ = { ( ξ ik ) k ∈ Ki | i ∈ I } and B b ξ = { ( η jλ ) λ ∈ Λ j | j ∈ J } . (26)We prove that there exists a δ -cut ξ in ( X, d ) such that ( ξ a ) a ∈ A converges to ξ .We define ξ = ( A ξ , B ξ ) , where A ξ = { ( x σ ) σ ∈ Σ | ( x σ ) σ ∈ Σ is a right d S − Cauchy net in (X , d) such that ( φ (x σ ) σ ∈ Σ ∈ A b ξ } (27)and B ξ = { ( y ρ ) ρ ∈ P | ( y ρ ) ρ ∈ P is a left d S − Cauchy net in (X , d) such that ( φ (y ρ ) ρ ∈ P ∈ B b ξ } . (28)By Propositions 3.16 and 3.18 the classes A ξ and B ξ are non-void. We first verify that ξ = ( A ξ , B ξ ) constitute a δ -cutin ( X, d ) . For this we need to show that the pair ( A ξ , B ξ ) satisfies the conditions of Definition 3.6. We first prove thevalidity of Condition (i) of Definition 3.6. Let ( x σ ) σ ∈ Σ ∈ A ξ and ( y ρ ) ρ ∈ P ∈ B ξ . Then, by construction of A ξ and B ξ ,we have lim ρ,σ b d ( φ ( y ρ ) , φ ( x σ )) = 0 . Hence, Proposition 3.12 implies that lim σ,ρ d ( y σ , x ρ ) = 0 . To prove that ξ satisfiesthe second condition of Definition 3.6, let ( x σ ) σ ∈ Σ , ( x ρ ) ρ ∈ P be two right d S -Cauchy nets of A ξ . Since ( φ ( x σ )) σ ∈ Σ , ( φ ( x ρ )) ρ ∈ P belong to A b ξ , it follows by the definition of ξ that they are right b d -cofinal. Hence, Proposition 3.12 impliesthat ( x σ ) σ ∈ Σ and ( x ρ ) ρ ∈ P are right d -cofinal. Finally, condition (iii) of Definition 3.6 is an immediate consequence ofthe maximality of A ξ and B ξ , respectively.Similarly we can prove conditions (ii) and (iii) in the case of left d S -Cauchy nets, members of B ξ .We now prove that ( ξ a ) a ∈ A converges to ξ . Indeed, according to Proposition 3.16 there exists a right d S -Cauchy net ( x σ ) σ ∈ Σ in ( X, d ) such that the nets ( ξ a ) a ∈ A and ( φ ( x σ )) σ ∈ Σ are right b d -cofinal. By construction of ξ we have that ( x σ ) σ ∈ Σ ∈ A ξ . By Proposition 3.13 we have that φ ( x σ ) −→ ξ . Since ( φ ( x σ )) σ ∈ Σ and ( ξ a ) a ∈ A are right b d -cofinl,Proposition 3.7 implies that ξ a −→ ξ . It follows that ( X, d ) is δ -complete.11 PREPRINT - W
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Proposition 3.20.
A quasi-pseudometric space ( X, d ) is δ -complete if and only if ( X, U d ) is U d -complete. Proof.
It is an immediate consequence of Proposition 3.1.
Proposition 3.21.
Let ( X, d ) be a quasi-pseudometric space and let U d be the quasi-uniformity induced on X by d .Then, the space ( X, U d ) is U d -complete if and only if the space ( X, d ) is δ -complete. Proof.
The result is an immediate consequence of Proposition 3.1 and the fact that the familly U ǫ = { ( x, y ) ∈ X × X | d ( x, y ) < ǫ | , ǫ > } is a base for U d . Lemma 3.22.
Let ( X, U ) and ( Y, V ) be arbitrary quasi-uniform spaces and let f : ( X, U ) −→ ( X, V ) be a quasi-uniformly continuous mapping. If ξ = ( A ξ , B ξ ) where A ξ = { ( x i a ) a ∈ A | i ∈ I } and B ξ = { ( y j β ) β ∈ B | j ∈ J } is a U -cutin ( X, U ) , then f ( ξ ) = ( f ( A ξ ) , f ( B ξ )) where f ( A ξ ) = { ( f ( x i a )) a ∈ A | i ∈ I } and f ( B ξ ) = { ( f ( y j β )) β ∈ B | j ∈ J } is aa V -cut in ( X, V ) . Proof.
Since f is quasi-uniform continuous, if ( x a ) a ∈ A is a right U S -Cauchy net in ( X, U ) with a left U S -Cauchy net ( y β ) β ∈ B as conet, then ( f ( x a )) a ∈ A is a right V S -Cauchy net in ( Y, V ) with ( f ( y β )) β ∈ B a left V S -Cauchy conet. Therest is obvious.The following proposition is evident. Proposition 3.23.
A closed subspace of a U -complete quasi-uniform space ( X, U ) is U -complete. Proposition 3.24. If { ( X i , U i ) | i ∈ I } is a family of U i -complete quasi-uniform spaces, then the product space ( e X, e U ) = ( Y i ∈ I X i , Y i ∈ I U i ) is e U -complete. Proof.
Let (Ξ a ) a ∈ A be a e U -Cauchy net in ( e X, e U ) . Then, there exists a a e U -cut Ξ = ( A Ξ , B Ξ ) in ( e X, e U ) such that (Ξ a ) a ∈ A ∈ A Ξ . Let (Ξ a ) a ∈ A ∈ A Ξ and ( H β ) β ∈ B ∈ B Ξ . Let Ξ a = { Ξ i a | i ∈ I } for each a ∈ A and H β = { H j β | j ∈ J } for each β ∈ B . Fix an i ∈ I and choose a U i ∈ U i . Given U i × Y j = i ( X j × X j ) there are a ∈ A and β ∈ B such that:(1) ( x a i , x a ′ i ) ∈ U whenever a ≥ a , a ′ ≥ a and a ′ (cid:3) a ; (2) ( y β ′ j , y β j ) ∈ U whenever β ≥ β , β ′ ≥ β and β ′ (cid:3) β and (3) ( y β j , x a i ) ∈ U whenever a ≥ a and β ≥ β . Thus, Ξ i = ( A Ξ i , B Ξ i ) is a U i -cut in ( X i , U i ) and ( x ai ) a ∈ A is a U i -Cauchy net in ( X i , U i ) . Thus, ( x ai ) a ∈ A converges to a point x i ∈ X i . Let e x = { x i | i ∈ I } . Then, it is easy to verifythat x a −→ e x. The following proposition is evident.
Proposition 3.25.
A closed subspace of a U -complete quasi-uniform space ( X, U ) is U -complete. Proposition 3.26. (see[20, Theorem 1.7] Each quasi-uniform space ( X, U ) can be embedded in a product of quasi-pseudometric spaces. Theorem 3.27.
Any quasi-unifrom space ( X, U ) has a U -completion. Proof.
By Proposition 3.26, we have that ( X, U ) can be embedded in a product of quasi-pseudometric spaces Y i ∈ I ( X i , d i ) . Without loss of generality we may assume that d i ( x i , y i ) ≤ for all x i , y i ∈ X i and for all i ∈ I . ByTheorem 3.19 each space ( X i , d i ) has a ( d i ) S -completion ( b X i , b d i ) . Therefore, ( X, U ) can be embedded in Y i ∈ I ( b X i , b d i ) .That is, there exists a quasi-uniformly continuous mapping b φ : ( X, U ) −→ Y i ∈ I ( b X i , b d i ) such that b φ ( X ) ⊆ Y i ∈ I ( b X i , b d i ) .It follows from Propositions 3.23, 3.24 and 3.25 that b φ ( X ) ⊆ Y i ∈ I ( b X i , b d i ) , where b φ ( X ) is the closure of b φ ( X ) in Y i ∈ I ( b X i , b d i ) . Let b U = { U \ b φ ( X ) × b φ ( X ) | U is a member of the product quasi − uniformity for Y i ∈ I ( b X i , b d i ) } . (29)12 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020Then, by Propositions 3.23 and 3.26 we have that ( b X, b U ) is a b U -completion for ( X, U ) . In the present paper, we give a new completion for quasi-metric and quasi-uniform spaces which generalizes thecompletion theories of Doitchinov [8] and Stoltenberg [20]. The main contribution in this paper is the use of thenotion of the cuts of nets (sequences) which generalizes the idea of Doitchinov for a completeness theory for quasi-metric and quasi-uniform spaces. Doitchinov’s completeness theory for quiet spaces (a class of quasi-uniform spaces)is very well behaved and extends the completion theory of uniform spaces in a natural way. According to Doitchinov[8] a natural definition of a Cauchy net in a quasi-uniform space ( X, U ) (similar conditions must be satisfied for aquasi-metric space ( X, d ) ) has to be defined in such a manner that the following requirements are fulfilled:(i) Every convergent sequence is a Cauchy net;(ii) In the uniform case (i.e. when ( X, U ) the Cauchy nets are the usual ones.Further a standard construction of a completion ( b X, b U ) of any quasi-uniform space ( X, U ) should be possible suchthat:(iii) If ( X, U ) ⊆ ( Y, V ) , where the inclusions are understood as quasi-uniform (resp. quasi-metric) embeddings andthe second one is an extension of the former;(iv) In the case when ( X, U ) is a uniform space ( X ∗ , U ∗ ) is nothing but the usual uniform completion of ( X, U ) .As it is well known, in metric spaces the notions of completeness by sequences and by nets agree and, further, thecompleteness of a metric space is equivalent to the completeness of the associated uniform space. Therefore, a naturalrequirement for a satisfactory theory of completeness in quasi-metric spaces is the following:(v) In quasi-metric spaces the sequential completeness and completeness by nets agree.The following definitions is due to Doitchinov ((see [8, Condition Q and Definition 11])) and inspired us to define thenotion of cuts of nets. More precisely: Definition 4.1.
A quasi-uniform space ( X, U ) is called quiet provided that for each U ∈ U there exists V ∈ U suchthat, if x ′ , x ′′ ∈ X and ( x a ) a ∈ A and ( x β ) β ∈ B are two nets in X , then from ( x, x a ) ∈ V for a ∈ A , ( x β , y ) ∈ V for β ∈ B and ( x β , x a ) → it follows that ( x, y ) ∈ U . We say that V is Q - subordinated to U . Definition 4.2.
A net ( x a ) a ∈ A in a quasi-uniform space is D - Cauchy if there exists another net ( y β ) β ∈ B such that foreach U ∈ U , there exist a U ∈ A β U ∈ B satisfying ( y β , x a ) ∈ U whenever a ≥ a U ∈ A and β ≥ β U ∈ B . The space ( X, U ) is D - complete if every D -Cauchy net in X is convergent. Definition 4.3.
Two D -Cauchy nets ( x a ) a ∈ A and ( x β ) β ∈ B are called equivazent if every conet of ( x a ) a ∈ A is a conetof ( x β ) β ∈ B and vice versa. Proposition 4.1. (See [8, Proposition 12]). If two D -Cauchy nets in a quiet quasi-uniform space ( X, U ) have acommon conet, then they are equivalent.Î ´Zn the quasi-metric case the notions of D -completeness by nets is defined by sequences as follows: Definition 4.4.
A sequence ( x n ) n ∈ N in the quasi-metric space ( X, d ) is called D - Cauchy sequence provided that forany natural number k there exist a ( y k ) k ∈ N and an N k such that d ( y k , x a ) < k whenever m, n > N k .The concept of D -Cauchy net (sequence) proposed by Doitchinov enables him to realize the program outlined aboveunder the assumption of quietness.Moreover, D -completeness satisfies requirement (v). The following definition shows this fact. Proposition 4.2. (See [8, Theorem 9]). In the balanced quasi-metric spaces the notions of D -completeness by se-quences and by nets agree. Proof.
If a quasi-metric space is D -complete, then each D -Cauchy net converges in X , and thus each D -Cauchysequence converges in X . Conversely, we prove that the sequential D -completeness implies that every D -Cauchy netin X is convergent. Let ( X, d ) be a balanced quasi-metric space in which any D -Cauchy sequence converges and let ( x a ) a ∈ A be a D -Cauchy net in ( X, d ) . Then, ( x a ) a ∈ A is a D -Cauchy net in the quasi-uniform space U d generated by d . By [8, Page 208], U d = { U ǫ | ǫ > } where U ǫ = { ( x, y ) ∈ X × X | d ( x, y ) < ǫ } is a quiet quasi-uniform space.13 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020Since ( x a ) a ∈ A is D -Cauchy there exists a net ( y β ) β ∈ B such that for any U ǫ ∈ U d there are a Uǫ ∈ A and β Uǫ ∈ B suchthat ( y β , x a ) ∈ U whenever a ≥ a Uǫ , β ≥ β Uǫ or equivalently lim a,β d ( y β , x a ) = 0 . Therefore, for each n ∈ N , thereexists a n ∈ A , β n ∈ B such that ( y βn , x a n ) ∈ U n = { ( x, y ) ∈ X × X | d ( x, y ) < n } . It follows that ( x a n ) n ∈ N is a D -Cauchy sequence and thus it converges to a x ∈ X . Therefore, lim β ( y β , x ) = 0 ([8, Lemma 15]). Let ǫ > . Then,since U d is quiet there exists ǫ ′ > such that U ǫ ′ is Q -subordinated to U ǫ . Let n ∈ N such that n < ǫ ′ . Then, from ( y βn , x a ) ∈ U ǫ ′ , ( x, x ) ∈ U ǫ ′ and lim β ( y β , x ) = 0 we conclude that ( x, x a ) ∈ U ǫ . It follows that ( x a ) a ∈ A convergesto x .By Proposition 4.1, to each D -Cauchy net ( x a ) a ∈ A , it corresponds a set of pairs of nets-conets which lead to the notionof cut of nets, that is, a pair ( C , D ) where C contains all equivalent nets of ( x a ) a ∈ A and D contains all of their conets.The notion of U -cut defined in this paper is a cut of nets ( A , B ) where the members of A contains right U S -Cauchynets and B contains left U S -Cauchy nets as they are defined by Stoltenberg.At this point it is routine to check that all the requirements (i)-(iv) are satisfied for the proposed U -completion. Therequirement (v) is also satisfied as shows Proposition 3.17.The validity of requirement (v) usually does not hold for some completion theories. As remarked Stoltenberg [20,Example 2.4], there exists a sequentially right K -complete quasi-metric space ( X, d ) which is not right U S -complete.Actually, Stoltenberg [20, Theorem 2.5] proved that the equivalence holds for a more general definition of a right U S -Cauchy net. More precisely, Stoltenberg [20] gives a more general definition of a right U S -Cauchy net than thatwe use in this paper as follows: A net ( x a ) a ∈ A in a quasi-unifrom space ( X, U ) is called right (resp. left ) U S - Cauchy if for each U ∈ U there is a U ∈ A such that ( x β , x α ) ∈ U (resp. ( x α , x β ) ∈ U ) whenever α, β ∈ A and a ≥ β ≥ a U .However, using this definition one can find a quasi-metric space ( X, d ) which is left K -sequentially complete but not U S -complete. He support this claim by offering the following example [20, Example 2.4]: Let A be the family of allcountable subsets of the closed interval [0 , ] and let for every A ∈ A and k ∈ N , X Ak +1 = A ∪ { , , ..., k − k } and X A ∞ = [ k ≥ X Ak .Define X A = { X Ak | k = 1 , , ..., ∞} and J = { S {X A | A ∈ A} .Define d : J × J → R by d ( X Ak , X Bj ) = j if X Bj ⊂ X Ak , k = 1 , , ..., ∞ , j = 1 , , ..., . d ( X Ak , X Ak ) = 0 and d ( X Ak , X Bj ) = 0 otherwise.Stoltenberg proves that ( J , d ) is left K -sequentially complete and not d S -complete. To overcome the weaknesses ofthis definition, in order to develop his theory of U S -completeness, he uses Definition 3.1. In order to prove his mainresult of the construction of U S -completion he uses Theorem 2.5 which says that: A quasi-metric space ( X, d ) is d S -complete if and only if every left K -Cauchy sequence in ( X, d ) converges with respect to τ d in X . On the otherhand, the notion of d S -completeness plays a central role in the constructed U S -completion, since this completion isa subset of Y i ∈ I ( b X i , b d i ) ( ( b X i , b d i ) is the ( d i ) S -completion of the T quasi-pseudometric space ( X i , d i ) ). Gregori andFerrer [11], using Example 2.4 of [20] as a counter-example, showed that Stoltenberg’s result of Theorem 2.5, basedon his U S -Cauchy net ([20, Definition 2.1]), is not valid in general. In fact, the authors define a net Φ in ( J , d ) asfollows: Let D = N ∪ { a, b } , where N is the set of natural with the usual order and a, b / ∈ N , a = b and a ≥ k and b ≥ k for k ∈ N , a ≥ b , b ≥ a , a ≥ a and b ≥ b . Cleraly, D is a directed set. Then, Φ( k ) = X Ak for k ∈ N , Φ( a ) = X A ∞ , Φ( a ) = X B ∞ , where A, B ∈ A and A ⊂ B .Then, Φ is a right d S -Cauchy net. Indeed, let < ε < and λ ∈ N be such that λ < ε . We have k ≤ a , k ≤ b , k ≥ k λ ≤ a ≤ b and λ ≤ b ≤ a . The condition λ (cid:3) k can hold for some k, λ ∈ I , k, λ ≥ k in the followingcases:( a ) k, λ ∈ N , k, λ ≥ k , k > λ . 14 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020( b ) k = a , λ ∈ N , k, λ ≥ k .( c ) k = a , λ ∈ N , k, λ ≥ k .In the case ( a ), we have that X Aλ ⊂ X A k and thus d (Φ( k ) , Φ( λ )) = d ( x Ak , x A λ ) = 12 λ < k < ε .In the case ( b ), we have that we have that X Aλ ⊂ X A ∞ and thus d (Φ( a ) , Φ( λ )) = d ( x A ∞ , x A λ ) = 12 λ < k < ε .The case ( c ) is similar to ( c ).On the other hand, Φ is not convergent in ( J , d ) . Indeed, let X ∈ J \ X B ∞ . Then, for each k ∈ I , b ≥ k holds and thus d ( X, ϕ ( b )) = d ( X, X B ∞ ) = 1 .It concludes that for each ε > , X B ∞ / ∈ B ( X, ε ) holds. Therefore, for any ε > no final segment of Φ is containedin B ( X, ε ) which implies that Φ does not converge to X . Similarly, if X = X B ∞ , then a ≥ k for each k ∈ I and d ( X B ∞ , X A ∞ ) = 1 .Î ˚Uence the problem that arises here is that the net Φ , although non-convergent, is a U S -Cauchy net. At this point wewill look at Doitchinov’s view on this problem using another definition of cauchyness he used. More precisely: Definition 4.5. (See [7, Definition, Page 129]). Let ( X, d ) be a quasi-pseudometric space. A sequence ( x n ) n ∈ N iscalled Cauchy sequence if for every natural number k there are a y k ∈ X and an N k ∈ N such that d ( y k , x n ) < k when n > N k .According to Doitchinov a non-formal objection to this definition of Cauchyness is illustrated by the following exam-ple. Example 4.3. (Sorgenfrey line). Let R be the real line equipped with the quasi-metric d ( x, y ) = (cid:26) y − x if x ≤ y , if x > y .For each x ∈ X , the collection { [ x, x + r ) | r > } form a local base at the point x for the topology generated by d in X .According to Doitchinov [7, Page 130]: The sequence ( − n ) n ∈ N , although nonconvergent, is a Cauchy sequence inthe sense of Definition 4.5. However, in view of the special character of the topology on the space (R,d), it seems veryinconvenient to regard this sequence as a potentially convergent one, i.e. as one that could be made convergent bycompleting the space. As we describe above, in order to avoid this unwanted phenomenon, Doitchinov has been introduced the D -completeness (see Definition 4.2).On the other hand, Gregori and Ferrer [11] also proposed a new definition of a right U S -Cauchy net, for which theequivalence to sequential completeness holds (requirement (v)). Definition 4.6.
A net ( x a ) a ∈ A in a quasi-metric space ( X, d ) is called GF -Cauchy if one of the following conditionsholds:(i) For every maximal element a ∗ ∈ A the net ( x a ) a ∈ A converges to x a ∗ ;(ii) A has no maximal elements and ( x a ) a ∈ A converges in X ;(iii) A has no maximal elements and ( x a ) a ∈ A satisfies Definition 2.1 of [20].More recently Cobzas [5] has contributed new results in Stoltenberg’s completion theory. In order to avoid the short-comings of the preorder relation, as, for instance, those put in evidence by Example of Gregori and Ferrer, he proposes15 PREPRINT - W
EDNESDAY ND S EPTEMBER , 2020a new definition of right K -Cauchy net in a quasi-metric space for which the corresponding completeness is equivalentto the sequential completeness. Definition 4.7.
A net ( x a ) a ∈ A in a quasi-metric space ( X, d ) is called strongly Stoltenberg-Cauchy if for every ǫ > there exists a ǫ ∈ A such that, for all a, β ≥ a ǫ ( β ≤ a ∨ a ≁ b implies that d ( x a , x β ) < ǫ where a ∨ a ≁ b means that a, β are incomparable (that is, no one of the relations a ≤ β or β ≤ a holds).It is interesting to see how U -Competion will become if we change the members of the classes of a U -cuts with Gregoriand Ferrer Cauchy nets or Cobzas Cauchy nets respectively. References [1] Andrikopoulos A., Completeness in quasi-uniform spaces,
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