# A Solution to the Completion Problem for Quasi-Pseudometric Spaces

Authors: Athanasios Andrikopoulos
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

The different notions of Cauchy sequence and completeness proposed in the literature for quasi-pseudometric spaces do not provide a satisfactory theory of completeness and completion for all quasi-pseudometric spaces. In this paper, we introduce a notion of completeness which is classical in the sense that it is made up of equivalence classes of Cauchy sequences and constructs a completion for any givenT0 quasi-pseudometric space. This new completion theory extends the existing completion theory for metric spaces and satisfies the requirements posed by Doitchinov for a nice theory of completeness.


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## Body

## 1. Introduction

A quasi-pseudometric space(X,d) is a set X together with a nonnegative real-valued function d:X×X→ℝ (called a quasi-pseudometric) such that, for every x,y,z∈X, (i) d(x,x)=0 and (ii) d(x,y)≤d(x,z)+d(z,y). If d satisfies the additional condition (iii) d(x,y)=0 which implies that x=y, then d is called a quasimetric on X. A quasimetric is a metric provided that d(x,y)=d(y,x). The conjugate of a quasi-pseudometric d on X is the quasi-pseudometric d-1 given by d-1(x,y)=d(y,x). By ds we denote the (pseudo)metric given by ds=max{d(x,y),d-1(x,y)}. Each quasi-pseudometric d on X induces a topology τd on X which has as a base the family of d-balls {Bd(x,r):x∈X,r>0}, where Bd(x,r)={y∈X:d(x,y)<r}. A quasi-pseudometric space is T0 if its associated topology τd is T0. In that case axiom (i) and the T0-condition can be replaced by (i′)  ∀x,y∈X, d(x,y)=d(y,x)=0⇔x=y. It is well known that every metric space has an (up to isometry) unique metric completion. For quasi-pseudometric spaces, the notion of completeness presents serious difficulties. The problem has been approached by several authors in [1–4], and so forth, but the solutions proposed have not solved these difficulties. Since a T0 quasi-pseudometric space is a generalization of a metric space (the difference between pseudometrics and metrics is purely topological. In fact a pseudometric d on X is a metric if and only if the topology it generates is T0), any completion theory for T0 quasi-pseudometric spaces should generalize the usual completion theory for metric spaces. Traditionally this is done by generalizing the concept of Cauchy sequence and that of the convergence of a sequence. The appropriate generalization of the notion of Cauchy sequence in quasi-pseudometric spaces is no longer obvious. There are various generalizations to the notion of Cauchy sequence, but, up to now, none of these generalizations is able to give a satisfying completion theory for all quasi-pseudometric spaces. In fact, there exist many different notions of quasi-pseudometric completeness in the literature. More precisely, this problem has been studied in [3], where seven different notions of Cauchy sequence are presented. (The definitions are that of left and right  p-Cauchy sequence (p is the quasimetric) defined by Reilly [5] and Subrahmanyam [6], respectively, p-Cauchy sequence defined by Tan [7], right  K- and left  K-Cauchy sequence defined by Kelly [2], and weakly  right  K- and weakly  left  K-Cauchy sequence defined in [3].) By combining the seven notions of Cauchy sequences with the topologies τd, τd-1, and τds, we may reach a total of fourteen different definitions of “complete space” (considering the symmetry of using the d-1 instead of d). Since the concept of Cauchy sequence generalizes the concept of convergent sequence, it would be desirable that the notion of Cauchy sequence in any quasi-pseudometric space has to be defined in such a manner that (*) every convergent sequence is a Cauchy sequence. However, only one of the definitions presented in [3] satisfies the requirement (*). This definition is the following: a sequence (xn)n∈ℕ in a quasi-pseudometric space (X,d) is called Cauchy sequence, if for each σ∈ℕ there are a yσ∈X and a Nσ∈ℕ such that d(yσ,xn)<1/σ when n>Nσ. Unfortunately, this definition has two serious disadvantages. (i) The property of Cauchyness of a sequence depends not only on its terms but also on some other points which need not to belong to it (see [1, Example 2] and [2, page 88]). This allows sometimes for a convergent Cauchy sequence to cease Cauchyness if we remove the limit point from the space. (ii) Consider the real line equipped with the quasimetric d(x,y)=y-x, if x≤y and d(x,y)=1 if x>y. The sequence (-1/n)n∈ℕ is a Cauchy one in the sense of the above definition. However (in Doitchinov’s words), “in view of the special character of the Sorgenfrey’s topology, it seems very inconvenient to regard this sequence as a potentially convergent one.”Doitchinov in [1] introduced a class of quasimetric spaces, for which a satisfactory theory of completeness exists. According to Doitchinov, a notion of Cauchy sequence in any quasimetric space has to be defined in such a manner that this definition provides the properties that convergent sequences are Cauchy, and it agrees with the usual definition for metric spaces. Moreover, the suggested completion must be a monotone operator with respect to inclusion and give rise to the usual pseudometric completion in the pseudometric case. By means of Cauchy sequences which satisfy the above requirements, another concept of Cauchy sequence is proposed and a completion of a quasimetric space is constructed. However, this construction satisfies natural requirements only for balanced quasimetric spaces. (A quasimetric (X,d) is called balanced if d satisfies the following condition: If (xn)n∈ℕ and (xm)m∈ℕ are two sequence in (X,d) and x,y∈X, then from d(x,xn)≤r1 for each n, d(xm,y)≤r2 for each m, and limm,nd(xm,xn)=0 it follows that d(x,y)≤r1+r2.) Unfortunately, Doitchinov’s condition of balancedness turns out to be rather restrictive. For instance, balanced T0 quasimetrics induce completely regular Hausdorff topologies (see [1, Corollary 3] and [8, Page 208]) and totally bounded balanced T0 quasimetrics induce uniformities (see [9, 10]). Künzi and Kivuvu [11, 12] localize Doitchinov’s idea of balancedness; that is, they do not work with arbitrary Cauchy filter pairs but only with those that they call balanced Cauchy filter pairs. Before stating the proposed completion theory from Künzi and Kivuvu, we give some definitions from [11]. Let (X,d) be a T0-quasimetric space. It is known that (X,d) can be isometrically embedded into a T0-quasimetric space (X*,d*), where the associated metric space (X*,ds*) is (isometric to) the completion of the metric space (X,ds). We recall that (X,d) is called bicomplete provided that each ds-Cauchy filter converges in (X,ds). Let A,B be nonempty subsets of X. The 2-diameter from A to B is defined by Φd(A,B)=sup{d(a,b)∣a∈A,b∈B}. We will say that a pair (ℱ,𝒢) of filters ℱ and 𝒢 on X is a Cauchy filter pair on (X,d) if infF∈ℱ,G∈𝒢Φd(F,G)=0. An arbitrary Cauchy filter pair (ℱ,𝒢) on X is said to converge to x∈X provided that infG∈𝒢Φd(x,G)=0 and infF∈ℱΦd(F,y)=0. A Cauchy filter pair (ℱ,𝒢) on (X,d) is said to be balanced on (X,d) if for each x,y∈X we have d(x,y)≤infG∈𝒢Φd(x,G)+infF∈ℱΦd(F,y). The space (X,d) is called B-complete provided that each balanced Cauchy filter pair converges in X. Let X+ be the set of all balanced Cauchy filter pairs on (X,d) and let (ℱ,𝒢),(ℱ′,𝒢′)∈X+. Then, the formula d+=((ℱ,𝒢),  (ℱ′,𝒢′))=infF∈ℱ,G′∈𝒢Φd(F,G′)=infF∈ℱ,G′∈𝒢supf∈F,g′∈G′d(f,g′) defines the distance from (ℱ,𝒢) to (ℱ′,𝒢′). The space (X+,d+) is a quasi-pseudometric space. We define an equivalence relation ≅ on the quasi-pseudometric space (X+,d+) as follows: (ℱ,𝒢)≅(ℱ′,𝒢′)((ℱ,𝒢),  (ℱ′,𝒢′)∈(X+,d+)) if and only if d+((ℱ,𝒢),  (ℱ′,𝒢′))=0 and d+((ℱ′,𝒢′),  (ℱ,𝒢))=0. Let X+^ be the set of all equivalence classes qX+^(ℱ), where ℱ∈X+, with respect to ≅. Then, d+^ on X+^ defined by d+^(qX+^(ℱ),  qX+^(𝒢))=d+(ℱ,𝒢) whenever ℱ,𝒢∈X+ determines a T0-quasimetric d+^ on X+^. According to [11, Lemma 5] we can identify (X+^,d+^) with the subspace (Xβ,dβ) of all balanced Cauchy filter pairs on (X,d) that are minimal elements in the space (X+,d+) of all balanced Cauchy filter pairs on (X,d). Künzi and Kivuvu in [11, Theorem 2] prove that if (X,d) is a quasi-pseudometric space then (Xβ,dβ) is B-complete. The space (Xβ,dβ) will be called the standardB-completion of (X,d). The authors also prove that the constructed standard B-completion extends the bicompletion of every T0-quasimetric space. In contrast to D-completion, the B-completion can be strictly larger than the bicompletion in totally bounded T0-quasimetric spaces (see [12, Example 8]). But, sometimes, the B-completion coincides with the bicompletion (see [11, Example 4]). In the present paper, we develop a nonsymmetric completion theory which gives a solution to the problem and extends naturally the existing completion theory for metric spaces. The following example shows that the proposed completion differs from the B-completion.Example 1 (see [2, Example 5.4]).
LetX={-1/(n+1),1/(n+1),n∈ℕ}. For each x,y∈X, let
(1)d(x,y)={1ifx<0<y,0ify≤x,min{1,∣x-y∣}otherwise.
It is easy to check that (X,d) is a T0-quasi-pseudometric space. Let (ℱ,𝒢) be the filter pair on X generated by 〈(1/(n+1))n∈ℕ,(-1/(n+1))n∈ℕ〉. We have that (ℱ,𝒢) is a Cauchy filter pair on (X,d), which is not balanced, since
(2)1=d(-14,14)≰infG∈𝒢Φd(-14,G)+infF∈ℱΦd(F,14)=14+14=12.
Therefore, theB-completion (Xβ,dβ) of (X,d) does not add new points to (X,d). In the proposed completion theory, the pair 〈(1/(n+1))n∈ℕ,(-1/(n+1))n∈ℕ〉 is a κ-cut in (X,d) (see Definition 8). This κ-cut defines a new point x0~ in a complete quasi-pseudometric space (X~,d~) (see Theorem 31). If the mapping ϕ~ is the isometric embedding (a map f:X→Y between quasi-pseudometric spaces is called an isometric embedding if d(f(x),f(y))=d(x,y) for all x,y∈X and it is called an isometry if it is a surjective isometric embedding) ϕ~:(X,d)→(X~,d~) such that ϕ~(X) is a dense subspace of (X~,d~), then (ϕ~(-(1/(n+1)))n∈ℕ and (ϕ~(1/(n+1)))n∈ℕ converge to x0~ with respect to τd~ and τd~-1, respectively.In this paper, (i) we propose a solution to the problem of quasi-pseudometric completion, calledκ-completion; and (ii) we show that the proposed definition of Cauchy sequence and the constructed completion satisfy the requirements posed by Doitchinov for a satisfactory theory of completion of quasi-pseudometric spaces. To do that, we introduce the notion of κ-cut which, in a sense, is a generalization of the notion of the equivalence class of Cauchy sequences in metric spaces. This notion is inspired by the notion of the MacNeille-Dedekind’s cut [13] (see also [14]). In order to construct a κ-cut, we make use of pairs “sequence-cosequence” as in the Doitchinov’s quasimetric completion. The notion of κ-cut (a) allows us to restrict the “area” where a new point must stand and (b) enables us to construct the κ-completion by using Cauchy sequences instead of κ-cuts.


## 2.κ-Cauchy Cut

Throughout the paper(X,d) will be an arbitrary quasi-pseudometric space, except the cases when it is explicitly stated that the space is T0. According to Doitchinov a sequence (yσ)σ∈ℕ is called a cosequence to (xn)n∈ℕ, if for any ɛ>0 there are σɛ,nɛ∈ℕ such that d(yσ,xn)<ɛ when σ≥σɛ and n≥nɛ. In this case, we write d(yσ,xn)→0 or limσ,nd(yσ,xn)=0. Generally, given two sequences (xn)n∈ℕ and (xm)m∈ℕ in (X,d), we write limn,md(xn,xm)=r if for any ɛ>0 there are nɛ,mɛ∈ℕ such that |d(xn,xm)-r|<ɛ when n≥nɛ and m≥mɛ.Definition 2 (see [3]).
A sequence(xn)n∈ℕ of a quasi-pseudometric space (X,d) is called right (resp. left) K- Cauchy, if for any ɛ>0 there is an nɛ∈ℕ such that d(xn,xn′)<ɛ for each n≥n′≥nɛ (resp. n′≥n≥nɛ). One calls nɛ (resp. xnɛ) the extreme index (resp. extreme point) of (xn)n∈ℕ for ɛ.Definition 3.
Let(X,d) be a quasi-pseudometric space and let (xn)n∈ℕ, (xm)m∈ℕ be two sequences on it. One says that (xn)n∈ℕ is right (resp. left) d-cofinal to  (xm)m∈ℕ, if for each ɛ>0 there exists nɛ∈ℕ satisfying the following property: for each n≥nɛ there exists mn∈ℕ such that d(xm,xn)<ɛ (resp. d(xn,xm)<ɛ) whenever m≥mn. The sequences (xn)n∈ℕ and (xm)m∈ℕ are right (resp. left) d-cofinal if (xn)n∈ℕ is right (resp. left) d-cofinal to (xm)m∈ℕ and vice versa.Proposition 4.
Let(xn)n∈ℕ be a right K-Cauchy (left K-Cauchy) sequence in a quasi-pseudometric space (X,d) with a subsequence (xnζ)nζ∈ℕ. Then, (xn)n∈ℕ and (xnζ)nζ∈ℕ are right (resp. left) d-cofinal.Proof.
Let(xn)n∈ℕ and (xnζ)nζ∈ℕ be as in the statement of the proposition. Let ɛ>0 be given. We know that from some index nɛ onwards we have that n≥n′≥nɛ implies that d(xn,xn′)<ɛ/2. Now if n≥nɛ we can find Nζ∈ℕ such that Nζ>n (as the nζ form a strictly increasing sequence of integers). Hence, for each nζ≥Nζ we have d(xnζ,xn)<d(xnζ,xNζ)+d(xNζ,xn)<ɛ/2+ɛ/2=ɛ. Hence, (xn)n∈ℕ is right d-cofinal to (xnζ)nζ∈ℕ. The converse is similar.Proposition 5.
In every quasi-pseudometric space(X,d), two right d-cofinal right K-Cauchy sequences have the same cosequences.Proof.
Let(xn)n∈ℕ and (xm)m∈ℕ be two right d-cofinal sequences. Suppose that (yσ)σ∈ℕ is a cosequence of (xm)m∈ℕ. Fix an ɛ>0. Then there exists σɛ, mɛ∈ℕ such that d(yσ,xm)<ɛ/2 for σ≥σɛ and m≥mɛ. On the other hand, there is nɛ∈ℕ with the following property: for each n≥nɛ there exists mn∈ℕ such that d(xm,xn)<ɛ/2 whenever m≥mn. If m(n)=max{mɛ,mn}, then d(yσ,xn)≤d(yσ,xm(n))+d(xm(n),xn)<ɛ for each σ≥σɛ, n≥nɛ. Hence, (yσ)σ∈ℕ is a cosequence of (xn)n∈ℕ.Similarly we can prove the following proposition.Proposition 6.
In every quasi-pseudometric space(X,d), two left d-cofinal left K-Cauchy sequences are cosequence of the same sequences.The two previous propositions imply the following corollary.Corollary 7.
In every quasi-pseudometric space(X,d), two right (left) d-cofinal sequences have the same limit points for d (resp. d-1).In the following definition, without loss of generality, we may suppose that, forɛ′≤ɛ, it is nɛ′≥nɛ.Definition 8.
Let(X,d) be a quasi-pseudometric space. We call κ-cut in X an ordered pair ξ=(𝒜,ℬ) of families of right K-Cauchy sequences and left K-Cauchy sequences, respectively, with the following properties.(i)
For any(xn)n∈ℕ∈𝒜 and any (yσ)σ∈ℕ∈ℬ there holds
(3)limσ,nd(yσ,xn)=0.(ii)
Any two members of the family𝒜 (resp. ℬ) are right (resp. left) d-cofinal.(iii)
The classes𝒜 and ℬ are maximal with respect to set inclusion.We call𝒜 (resp. ℬ)  first  (resp. second) class of ξ. In what follows, for simplicity of the proofs, we call the elements of ℬ left K-Cauchy consequences of the elements of 𝒜.Definition 9.
To everyx∈X one chooses a κ-cut ϕ(x)=(𝒜,ℬ), where 𝒜 consists of right K-Cauchy sequences which converges to x with respect to τd and ℬ consists of left K-Cauchy cosequences which converges to x with respect to τd-1. The sequence (x)=x,x,x,… itself belongs to both of the classes. If there are not right K-Cauchy sequences (resp. left K-Cauchy cosequences) converging to x, then, ϕ(x)=((x),ℬ) (resp. ϕ(x)=(𝒜,(x))). If x is an isolated point for τd and τd-1, respectively, then ϕ(x)=((x),(x)).As we can mapX to the set of all κ-cuts, it is possible that for some x∈X there correspond many κ-cuts; let ϕi(x), i∈Ix (the index set Ix depends on the choice of x). Then, we define
(4)Φ(x)={ϕi(x)∣i∈Ix}.
Without loss of generality, we assume that ϕ(x)=ϕi0(x) for some i0∈Ix.Remark 10.
If the space(X,d) is T0, then the function ϕ defined above is an injective function (one-to-one) of X into X^. Indeed, let x,y∈X be such that ϕ(x)=ϕ(y). Then, (x),(y)∈𝒜ϕ(x)∩ℬϕ(x)∩𝒜ϕ(y)∩ℬϕ(y). Thus, d(x,y)=d(y,x)=0 which implies that x=y.Definition 11.
One callsκ-Cauchy sequence any right K-Cauchy sequence which is member of the first class of a κ-cut.Definition 12.
Two rightK-Cauchy sequences (xn)n∈ℕ and (xm)m∈ℕ defined in a quasi-pseudometric space (X,d) are called κ-equivalent if every left K-Cauchy cosequence to (xn)n∈ℕ is a left K-Cauchy cosequence to (xm)m∈ℕ and vice versa.Clearly,κ-equivalence defines an equivalence relation on X.An immediate consequence of Definitions8, 11, and 12 is the following corollary.Corollary 13.
In every quasi-pseudometric space(X,d), two nonconstant κ-Cauchy sequences which belong to the same κ-cut are κ-equivalent.Remark 14.
In view of Corollary13, we have that the first class of a κ-cut is an equivalence class of the κ-Cauchy sequences that are considered to be equivalent by κ-equivalence relation.Definition 15.
A quasi-pseudometric space(X,d) is called κ-complete if every κ-Cauchy sequence converges.Definition 16.
Aκ-completion of a quasi-pseudometric (X,d) is a κ-complete quasi-pseudometric space (X^,d^) in which (X,d) can be isometrically embedded as a dense subspace.If aκ-cut ξ=(𝒜,ℬ) does not belong to ⋃x∈XΦ(x) we say that (𝒜,ℬ) is a κ-gap. The set of all κ-gaps of X is symbolized by X¯. Let us define
(5)X^=ϕ(X)⋃‍X¯.Throughout the paper, for everyξ∈X^, 𝒜ξ, ℬξ denote the two classes of ξ. In this case, we write ξ=(𝒜ξ,ℬξ).Definition 17.
Let(X,d) be a quasi-pseudometric space. Suppose that r is a nonnegative real number, ξ′,ξ′′∈X^, (xn′)n∈ℕ∈𝒜ξ′, and (xm′′)m∈ℕ∈𝒜ξ′′. We put d^(ξ′,ξ′′)≤r if(i)
𝒜ξ′=𝒜ξ′′ or(ii)
for eachɛ>0 there are nɛ′, mɛ′′∈ℕ such that
(6)d(xn′,xm′′)<r+ɛ when n≥nɛ′ and m≥mɛ′′. If ξ′=ϕ(x) for some x∈X, then the arbitrary sequence (xn′)n∈ℕ always coincides with the fixed sequence, for which xn′=x for all n∈ℕ. That is, d^(ϕ(x),ξ′′)≤r if
(7)d(x,xm′′)<r+ɛ
when m≥mɛ′′.Then we let(8)d^(ξ′,ξ′′)=inf{r∣d^(ξ′,ξ′′)≤r}.Proposition 18.
Letξ′,ξ′′∈X^, (xn′)n∈ℕ∈𝒜ξ′ and (xm′′)m∈ℕ∈𝒜ξ′′. Suppose that d^(ξ′,ξ′′)=0. Then, (xm′′)m∈ℕ is right d-cofinal to (xn′)n∈ℕ.Proof.
By the suppositions, for eachɛ>0 there are nɛ′, mɛ′′∈ℕ such that d(xn′,xm′′)<ɛ whenever n≥nɛ′ and m≥mɛ′′. The rest is evident by Definition 3.Proposition 19.
The truth ofd^(ξ′,ξ′′)≤r in Definition 17 depends only on ξ′, ξ′′, and r; it does not depend on the choice of the sequences (xn′)n∈ℕ and (xm′′)m∈ℕ.Proof.
Letξ′,ξ′′, (xn′)n∈ℕ, (xm′′)m∈ℕ, and r be as in the supposition of Definition 17. Further, let d^(ξ′,ξ′′)≤r. Then, for each ɛ>0 there are nɛ′, mɛ′′∈ℕ such that
(9)d(xn′,xm′′)<r+ɛ3
when n≥nɛ′ and m≥mɛ′′.
We choose two arbitrary sequences(xs′)s∈ℕ∈𝒜ξ′ and (xt′′)t∈ℕ∈𝒜ξ′′. Since (xn′)n∈ℕ is right d-cofinal to (xs′)t∈ℕ, there is Nɛ′∈ℕ satisfying the following property: for each n≥Nɛ′ there exists sn∈ℕ such that
(10)d(xs′,xn′)<ɛ3
whenever s≥sn. Let n~ɛ′=max{nɛ′,Nɛ′}. Then, we have(11)d(xs′,xm′′)≤d(xs′,xn~ɛ′)+d(xn~ɛ′,xm′′)<r+ɛ3+ɛ3=r+2ɛ3
whenever s≥sn~ɛ′ and m≥mɛ′′.
Analogously, since(xt′′)t∈ℕ is right d-cofinal to (xm′′)m∈ℕ, there is tɛ∈ℕ satisfying the following property: for each t≥tɛ there exists mt∈ℕ such that
(12)d(xm′′,xt′′)<ɛ3
whenever m≥mt. Let m(t,ɛ)=max{mɛ′′,mt}. Then, by (11) we have
(13)d(xs′,xt′′)≤d(xs′,xm(t,ɛ)′′)+d(xm(t,ɛ)′′,xt′′)<r+2ɛ3+ɛ3=r+ɛ
when s≥sn~ɛ′ and t≥tɛ.Proposition 20.
Letξ′,ξ′′∈X^, (xn′)n∈ℕ∈𝒜ξ′, and (xm′′)m∈ℕ∈𝒜ξ′′. Then,
(14)d^(ξ′,ξ′′)=limn,md(xn′,xm′′).Proof.
Letd^(ξ′,ξ′′)=r. Then, for any ɛ>0 there are nɛ′, mɛ′′∈ℕ such that
(15)d(xn′,xm′′)<r+ɛ
when n≥nɛ′ and m≥mɛ′′. To prove that r-ɛ<d(xn′,xm′′) for n≥nɛ′ and m≥mɛ′′, suppose to the contrary that there exist a subsequence (xnζ′)nζ∈ℕ of (xn′)n∈ℕ and a subsequence (xmυ′′)mυ∈ℕ of (xm′′)m∈ℕ such that for all xnζ′,xmυ′′ there holds d(xnζ′,xmυ′′)≤r-ɛ. Then, since Propositions 4, 5, and 6 imply that (xnζ′)nζ∈ℕ∈𝒜ξ′ and (xmυ′′)mυ∈ℕ∈𝒜ξ′′, it follows, according to Definition 17, that d^(ξ′,ξ′′)≤r-ɛ, a contradiction. Therefore, we have limn,md(xn′,xm′′)=r.Proposition 21.
d
^ is a quasi-pseudometric.Proof.
From Definition17 it follows immediately that d^(ξ,ξ)=0 and d^(ξ,ξ′)≥0 for all ξ,ξ′∈X^. To prove that d satisfies the triangle inequality let ξ,ξ′,ξ′′∈X^. We distinguish the following cases.(i)
𝒜ξ≠𝒜ξ′ and 𝒜ξ′≠𝒜ξ′′. Suppose that d^(ξ,ξ′)=r1, d^(ξ′,ξ′′)=r2, (xn)n∈ℕ∈𝒜ξ, (xm′)m∈ℕ∈𝒜ξ′, and (xs′′)s∈ℕ∈𝒜ξ′′. Then, for any ɛ>0 there are nɛ, mɛ′>0 such that d(xn,xm′)<r1+ɛ/2 whenever n≥nɛ and m≥mɛ′. Similarly, there are m~ɛ′, sɛ′′>0 such that d(xm′,xs′′)<r2+ɛ/2 whenever m≥m~ɛ′ and s≥sɛ′′. Let M0=max{mɛ,m~ɛ′}. Then, d(xn,xs′′)≤d(xn,xM0′)+d(xM0′,xs′′)<r1+r2+ɛ for each n≥nɛ, s≥sɛ′′. Hence, according to Definition 17, we have
(16)d^(ξ,ξ′′)≤r1+r2=d^(ξ,ξ′)+d^(ξ′,ξ′′).(ii)
𝒜ξ≠𝒜ξ′ and 𝒜ξ′=𝒜ξ′′. Suppose that d^(ξ,ξ′)=r. Since 𝒜ξ′=𝒜ξ′′, Definition 17 implies that d^(ξ,ξ′)≤r and d^(ξ′,ξ′′)=0. Hence
(17)d^(ξ,ξ′′)≤r=d^(ξ,ξ′)+d^(ξ′,ξ′′).(iii)
𝒜ξ=𝒜ξ′ and 𝒜ξ′≠𝒜ξ′′. This case is proved similar to the previous one.(iv)
𝒜ξ=𝒜ξ′ and 𝒜ξ′=𝒜ξ′′. This case is trivial.After the definition of the quasi-pseudometricd^, we are going to show that the function ϕ of Definition 9 is an isometric embedding of (X,d) into (X^,d^).Proposition 22.
For anyx,y∈X there holds d^(ϕ(x),ϕ(y))=d(x,y).Proof.
The assertion of the proposition follows from the fact thatx∈𝒜ϕ(x), y∈𝒜ϕ(y), and Proposition 20.Proposition 23.
For anyξ∈X^,(a)
if(xn)n∈ℕ∈𝒜ξ then d^(ξ,ϕ(xn))→0;(b)
if(yσ)σ∈ℕ∈ℬξ then d^(ϕ(yσ),ξ)→0.Proof.
(a)  Letɛ>0. Then, there exists nɛ>0 such that d(xn,xn′)<ɛ/2 whenever n≥n′≥nɛ. Now pick a right K-Cauchy sequence (xm)m∈N of 𝒜ξ. Since (xn)n∈ℕ is right d-cofinal to (xm)m∈ℕ, there is nɛ*∈ℕ satisfying the following property: for each n≥nɛ* there exists mn∈ℕ such that
(18)d(xm,xn)<ɛ2
whenever m≥mn. Suppose that n~ɛ=max{nɛ,nɛ*}. Fix an n≥n~ɛ and let (xs)s∈ℕ∈𝒜ϕ(xn). Then, xs→xn and so there exists an sɛ∈ℕ such that d(xn,xs)<ɛ/2 for each s≥sɛ. Hence,
(19)d(xm,xs)≤d(xm,xn)+d(xn,xs)<ɛ2+ɛ2=ɛ
for m≥mn and s≥sɛ. Since (xm)m∈ℕ∈𝒜ξ and (xs)s∈ℕ is an arbitrary right K-Cauchy sequence of 𝒜ϕ(xn), we conclude that d^(ξ,ϕ(xn))<ɛ. Hence, d^(ξ,ϕ(xn))<ɛ whenever n≥n~ɛ.
(b) The proof is similar to that of (a).Proposition23 implies the following corollary.Corollary 24.
The setϕ(X) is dense in (X^,d^).Proposition 25.
Let(ξγ)γ∈ℕ be a nonconstant right K-Cauchy sequence of (X^,d^) without last element. Then there exists a right K-Cauchy sequence (xn)n∈ℕ of (X,d) such that the sequences (ξγ)γ∈ℕ and (ϕ(xn))n∈ℕ are right d^-cofinal right K-Cauchy sequences.Proof.
(a)  The Construction of the Right  K-Cauchy Sequence  (xn)n∈ℕ  of  X. Let (ξγ)γ∈ℕ be a κ-Cauchy sequence in X^. Without loss of generality we can assume that, for each γ∈ℕ, 𝒜γ≠𝒜γ+1. Let n be a nonzero fixed natural number and suppose that γ*(n) is the smallest integer γ(n) with the property
(20)d^(ξγ,ξγ′)<13nforγ≥γ′≥γ(n).
Then, there holds
(21)γ*(n)≥γ*(n′)iffn′≥n.
For eachγ∈ℕ, we also fix a sequence
(22)(xn(γ))n(γ)∈ℕ∈𝒜ξγ.
If ξγ=ϕ(x) for some x∈X, then xn(γ)=x for each n(γ)∈ℕ. Let
(23)Λ(n)={γ*(n)+2ifγ*(n)iseven,γ*(n)+3ifγ*(n)isodd.
Then, for each λ≥Λ(n)/2 we have
(24)d^(ξ2λ,ξ2λ-1)<13n.
Hence, since 𝒜2λ-1≠𝒜2λ, there are n0(2λ)=n0(γ*(n),2λ)∈ℕ and n0(2λ-1)∈ℕ such that
(25)d(xn(2λ),xn(2λ-1))<13n
whenever n(2λ)≥n0(γ*(n),2λ) and n(2λ-1)≥n0(2λ-1). On the other hand, since (xn(2λ))n(2λ)∈ℕ is a right K-Cauchy sequence, there exists an n1(2λ)∈ℕ such that
(26)d(xn(2λ),xn′(2λ))<13n
for each n(2λ)≥n′(2λ)≥n1(2λ).
We putn~(γ*(n),2λ)=max{n0(γ*(n),2λ),n1(2λ)}. Then, we have
(27)n~(γ*(n),2λ)≥n~(γ*(n′),2λ)iffγ*(n)≥γ*(n′).
Clearly, (25) and (26) are valid if we replace n0(γ*(n),2λ) and n1(2λ), respectively, by n~(γ*(n),2λ). Let λ,λ′∈ℕ be such that Λ(n)/2≤λ′<λ. Then, since γ*(n)≤2λ′<2λ-1, similarly to the above we get
(28)d(xn(2λ-1),xn(2λ′))<13n
whenever n(2λ-1)≥n1(2λ-1) and n(2λ′)≥n2(2λ′) for some n1(2λ-1),n2(2λ′)∈ℕ. Let n~(2λ-1)=max{n0(2λ-1,n1(2λ-1)}. Then, by putting in (25), n(2λ)=n~(γ*(n),2λ), and n(2λ-1)=n~(2λ-1), we obtain that
(29)d(xn~(γ*(n),2λ),xn~(2λ-1))<13n.
On the other hand (28) implies that
(30)d(xn~(2λ-1),xn(2λ′))<13n
for each n(2λ′)≥n2(2λ′) for some n2(2λ′)∈ℕ. Hence,
(31)d(xn~(2λ-1),xn2(2λ′))<13n.
There are two cases to consider: (i) n~(γ*(n),2λ′)≥n2(2λ′); (ii) n2(2λ′)≥n~(γ*(n),2λ′).
In case (i), (28) implies that
(32)d(xn~(2λ-1),xn~(γ*(n),2λ′))<13n.
Hence,
(33)d(xn~(γ*(n),2λ),xn~(γ*(n),2λ′))<d(xn~(γ*(n),2λ),xn~(2λ-1))+d(xn~(2λ-1),xn~(γ*(n),2λ′))<13n+13n<1n.
In case (ii), we have n2(2λ′)≥n~(γ*(n),2λ′)=max{n0(γ*(n),2λ′),n1(2λ′)}; hence (26) implies that
(34)d(xn2(2λ′),xn~(γ*(n),2λ′))<13n.
Therefore, (29), (30), and (34) imply that
(35)d(xn~(γ*(n),2λ),xn~(γ*(n),2λ′))<d(xn~(γ*(n),2λ),xn~(2λ-1))+d(xn~(2λ-1),xn2(2λ′))+d(xn2(2λ′),xn~(γ*(n),2λ′))<13n+13n+13n<1n.
Hence, in any case, we have
(36)d(xn~(γ*(n),2λ),xn~(γ*(n),2λ′))<1n
for each λ≥λ′≥Λ(n)/2.
We consider the set(37)Θ={n~(γ*(n),2λ)∣n,λ∈ℕ}.
According to (27) we give to Θ the following order:
(38)ϑ′=n~(γ*(n′),2λ′)≾ϑ=n~(γ*(n),2λ)
if and only if
(39)λ=λ′,γ*(n′)≤γ*(n),orλ′<λ.
Since Θ is countable and ≾ is a linear order, without loss of generality, we may assume that Θ⊆ℕ and ≾⊆≤, where ≤ is the standard ordering of the natural numbers (this means that ≤ is a linear refinement of ≾). We now prove that the sequence
(40)(xn~(γ*(n),2λ))n~(γ*(n),2λ)∈ℕ
is a right K-Cauchy sequence which satisfies the assumptions of the lemma.
Letɛ>0 and nɛ=min{n∣2/n<ɛ}. Suppose that
(41)n~(γ*(nɛ),Λ(nɛ))≤n~(γ*(n′),2λ′)≤n~(γ*(n),2λ).
We prove that
(42)d(xn~(γ*(n),2λ),xn~(γ*(n′),2λ′))<ɛ.
If λ′=λ and γ*(n′)=γ*(n)), then n~(γ*(n),2λ)=n~(γ*(n′),2λ′). Therefore,
(43)d(xn~(γ*(n),2λ),xn~(γ*(n′),2λ′))=0<ɛ.
Suppose that n~(γ*(nɛ),Λ(nɛ))≤n~(γ*(n′),2λ′)<n~(γ*(n),2λ). Then, we have two cases to consider: (I) λ′=λ and γ*(n′)<γ*(n); (II) λ′<λ. In case (I), we have n~(γ*(nɛ),Λ(nɛ))≤n~(γ*(n′),2λ)<n~(γ*(n),2λ). Since (xn(2λ))n(2λ)∈ℕ is a right K-Cauchy sequence we have that
(44)d(xn~(γ*(n),2λ),xn~(γ*(n′),2λ′))=d(xn~(γ*(n),2λ),xn~(γ*(n′),2λ))<13nɛ<2nɛ<ɛ.
We proceed to prove case (II). In this case we have two subcases, (II1)γ*(n′)<γ*(n) and (II2)γ*(n)≤γ*(n′). In subcase (II1) we have n~(γ*(nɛ),Λ(nɛ))≤n~(γ*(n′),2λ′)<n~(γ*(n),2λ′). Since (xn(2λ′))n(2λ′)∈ℕ is a right K-Cauchy sequence we have that
(45)d(xn~(γ*(n),2λ′),xn~(γ*(n′),2λ′))<13nɛ.
Then, (36) and (45) imply that
(46)d(xn~(γ*(n),2λ),xn~(γ*(n′),2λ′))<d(xn~(γ*(n),2λ),xn~(γ*(n),2λ′))+d(xn~(γ*(n),2λ′),xn~(γ*(n′),2λ′))<1nɛ+13nɛ<2nɛ<ɛ.
In order to prove subcase (II2), we combine (29) and (30). So, we get
(47)d(xn~(γ*(n),2λ),xn(2λ′))<13nɛ+13nɛ=23nɛ
for each n(2λ′)≥n2(2λ′) for some n2(2λ′)∈ℕ. Now, we have two subcases to consider: the first subcase is that n~(γ*(n′),2λ′)≥n2(2λ′) and the second one is that n2(2λ′)≥n~(γ*(n′),2λ′)≥n~(γ*(n),2λ′)≥n~(γ*(nɛ),Λ(nɛ)). In the first subcase, (45) implies that
(48)d(xn~(γ*(n),2λ),xn~(γ*(n′),2λ))<23nɛ<2nɛ<ɛ.
In the second subcase, we have
(49)d(xn2(2λ′),xn~(γ*(n′),2λ′))<13nɛ.
Therefore, (45) and (49) imply that
(50)d(xn~(γ*(n),2λ),xn~(γ*(n′),2λ′))<d(xn~(γ*(n),2λ),xn2(2λ′))+d(xn2(2λ′),xn~(γ*(n′),2λ′))<23nɛ+13nɛ<2nɛ<ɛ.
Hence, in any case the inequality (42) is invalid.
(b)  (ϕ(xn~(γ*(n),2λ)))n~(γ*(n),2λ)∈ℕ  and  (ξγ)γ∈ℕ  Are Right  d^-Cofinal. We first show that (ξγ)γ∈ℕ is right d^-cofinal to (ϕ(xn~(γ*(n),2λ)))n~(γ*(n),2λ)∈ℕ. Fix an ɛ>0 and let Nɛ=min{n∣1/3n<ɛ}. Suppose that γ≥Λ(Nɛ). Then, we define (n~(γ*(n),2λ))γ=(n~(γ*(Nɛ),2λγ), where 2λγ=γ+mod(γ,2)+2.
We prove that(51)d(ϕ(xn~(γ*(n),2λ)),ξγ)<ɛ
for each n~(γ*(n),2λ)≥(n~(γ*(Nɛ),2λγ). That is, if (xs)s∈ℕ∈𝒜ξγ, then
(52)d(xn~(γ*(n),2λ),xs)<ɛ
for each s≥sɛ for some sɛ∈ℕ. Indeed, fix a λ∈ℕ such that Λ(Nɛ)<γ<2λγ<2λ-1<2λ. Then, from (25) which is valid for n(2λ)=n~(γ*(n),2λ) we have
(53)(xn~(γ*(n),2λ),xn(2λ-1))<13Nɛ
for each n(2λ-1)≥n0(2λ-1). Now, since Λ(Nɛ)<γ<2λ-1 we have
(54)(xn(2λ-1),xs)<13Nɛ
for some n(2λ-1), s onwards. On the other hand, since (xs)s∈ℕ is right K-Cauchy sequence we have
(55)(xs,xs′)<13Nɛ
for s≥s′≥sɛ. Finally, by combining the last three inequalities, we obtain
(56)d(xn~(γ*(n),2λ),xs)<13Nɛ+13Nɛ+13Nɛ<1Nɛ<ɛ
for each s≥sɛ. The last conclusion confirms the truth of (51).
To prove that(ξγ)γ∈ℕ is right d^-cofinal to ϕ(xn~(γ*(n),2λ))n~(γ*(n),2λ)∈ℕ, fix an ɛ>0. We define nɛ=min{n∣2/n<ɛ}. Now, let n~(γ*(n),2λ)≥n~(γ*(nɛ),Λ(nɛ)) and let γn~(γ*(n),2λ)=2λ+2. We prove that
(57)d^(ξγ,ϕ(xn~(γ*(n),2λ)))<13n
for each γ>γn~(γ*(n),2λ). Indeed, let (xs)s∈ℕ∈𝒜ξγ and (xt)t∈ℕ∈𝒜ϕ(n~(γ*(n),2λ)). Since
(58)d^(ξγ,ξ2λ-1)<13nɛ,d^(ξ2λ-1,ξ2λ)<13nɛ
using the same logic as above, we conclude that
(59)d(xs,xn~(γ*(n),2λ))<1nɛ
for each s≥sɛ for some sɛ∈ℕ. On the other hand, since xt→xn~(γ*(n),2λ) we have
(60)d(xn~(γ*(n),2λ),xt)<1nɛ
whenever t≥tɛ for some tɛ∈ℕ. Therefore, (59) and (60) imply that
(61)d(xs,xt)<d(xs,xn~(γ*(n),2λ))+d(xn~(γ*(n),2λ),xt)<2nɛ<ɛ
for each s≥sɛ and each t≥tɛ. Finally, (61) implies (57) and therefore the required sequence (xn)n∈ℕ of the hypothesis is the sequence (xn~(γ*(n),2λ))n~(γ*(n),2λ)∈ℕ.Proposition 26.
Let(η)=(ηδ)δ∈ℕ be a nonconstant left K-Cauchy sequence of (X^,d^) without last element. Then there exists a left K-Cauchy sequence (yσ)σ∈ℕ of (X,d) such that the sequences (ηδ)δ∈ℕ and (ϕ(yσ))σ∈ℕ are left d^-cofinal.Theorem 27.
Every quasi-pseudometric space has aκ-completion.Proof.
Let(X,d) be a quasi-pseudometric space and let (ξγ)γ∈ℕ be a κ-Cauchy sequence in the space (X^,d^). Then, by Definition 11, there exists a κ-cut ξ^=(𝒜ξ^,ℬξ^)∈X^ such that (ξγ)γ∈ℕ∈𝒜ξ^. Suppose that 𝒜ξ^={(ξγ(i)i)γ(i)∈ℕ∣i∈I} and ℬξ^={(ηδ(j)j)δ(j)∈ℕ∣j∈J}. Then, (ξγ)γ∈ℕ=(ξγ(i0)i0)γ(i0)∈ℕ for some i0∈I. We prove that there exists a κ-cut ξ in (X,d) such that (ξγ)γ∈ℕ converges to ξ with respect to τ(d^).
We define that𝒜ξ={(xn)n∈ℕ∣(xn)n∈ℕ is a right K-Cauchy sequence of X such that (ϕ(xn)n∈ℕ∈𝒜ξ^} and ℬξ={(yσ)σ∈ℕ∣(yσ)σ∈ℕ is a left K-Cauchy sequence of X such that (ϕ(yσ)σ∈ℕ∈ℬξ^}.
We first verify thatξ=(𝒜ξ,ℬξ) constitute a κ-cut in (X,d). For this we need to show that (A)
the classes𝒜ξ, ℬξ are nonvoid,(B)
the pair(𝒜ξ,ℬξ) satisfies the conditions of Definition 8.
We first show (A). We have two cases to consider.
Case A.1. There exists  i~∈I  such that the sequence  (ξγ(i~)i~)γ(i~)∈ℕ  is nonconstant. In this case, Proposition 25 implies that there is a right K-Cauchy sequence (xn)n∈ℕ of X such that (ξγ(i~)i~)γ(i~)∈ℕ and (ϕ(xn))n∈ℕ are right d^-cofinal. Therefore, (ϕ(xn))n∈ℕ∈𝒜ξ^ which implies that (xn)n∈ℕ∈𝒜ξ.
Case A.2. For each  i∈I, there exists  γ^(i)∈ℕ  such that  ξγ(i)i=ξγ(i)+1i  whenever  γ(i)≥γ^(i). Fix an i*∈I. Suppose that (xn)n∈ℕ∈𝒜ξγ^(i*)i*. Then, from limδ(j),γ(i*)d^(ηδ(j)j,ξγ(i*)i*)=0,  d^(ξγ(i*)i*,ξγ^(i*)i*)=0, and limnd^(ξγ^(i*)i*, ϕ(xn))=0, we conclude that limδ(j),nd^(ηδ(j)j,ϕ(xn))=0. It follows that (xn)n∈ℕ∈𝒜ξ. Therefore, in any case we have that 𝒜ξ≠∅. Similarly we prove that ℬξ≠∅.
To show (B), we first prove the validity of Condition (i) of Definition8. Let (xn)n∈ℕ∈𝒜ξ and (yσ)σ∈ℕ∈ℬξ. Then, by construction of 𝒜ξ and ℬξ, we have limσ,nd^(ϕ(yσ),ϕ(xn))=0. Hence, Proposition 22 implies that limσ,nd(yσ,xn)=0. To prove that ξ satisfies the second condition of Definition 8, let (xn)n∈ℕ and (xm)m∈ℕ be two right K-Cauchy sequences of 𝒜ξ. Since (ϕ(xn))n∈ℕ and (ϕ(xm))m∈ℕ belong to 𝒜ξ^, it follows by definition that they are right d^-cofinal. Hence, Proposition 22 implies that (xn)n∈ℕ and (xm)m∈ℕ are right d-cofinal. Clearly, the maximality of 𝒜ξ^ and ℬξ^ implies the maximality of 𝒜ξ and ℬξ, respectively. Hence, condition (iii) is satisfied.
It remains to prove that(ξγ)γ∈ℕ converges to ξ with respect to τ(d^). If (ξγ)γ∈ℕ is constant, then, ξγ=ξ for some γ~∈ℕ onwards. Hence, by the Definition 17, we have that limγd^(ξ,ξγ)=0. If the sequence (ξγ)γ∈ℕ is nonconstant, then there exists a right K-Cauchy sequence (xn)n∈ℕ∈𝒜ξ such that (ϕ(xn))t∈ℕ and (ξγ)γ∈ℕ are right d^-cofinal. Since (xn)n∈ℕ∈𝒜ξ, Proposition 23 implies that ϕ(xn)→ξ. So, by Corollary 7 we conclude that ξγ→ξ. This completes the proof.Definition 28.
Let(X,d) be a T0-quasi-pseudometric space. We define an equivalence relation ≈ on X^=X¯∪ϕ(X) as follows: ξ≈ξ′ if and only if 𝒜ξ=𝒜ξ′ whenever ξ,ξ′∈X¯ or ξ=ϕi(x) and ξ′=ϕj(x) for some i,j∈Ix whenever ξ,ξ′∈Φ(x), x∈X (see Definition 9). In the following ϕq:X^→X~ will denote the isometric quotient map with respect to ≈ previously defined. That is, ξ corresponds to ϕq(ξ)=Ξ whenever ξ∈X^. Clearly, ϕq is surjective. X~ will denote the set of all equivalence classes Ξ.Proposition 29.
Let(X,d) be a T0-quasi-pseudometric space and let X^, X~ be as above. Suppose that d~:X~×X~→ℝ is a function mapping defined by d~(Ξ,Ξ′)=d^(ξ,ξ′) whenever ξ,ξ′∈X^. Then, d~ determines a T0-quasi-pseudometric on X~.Proof.
Clearly,≈ is an equivalence relation on X. To prove that d~ is well-defined suppose that ξ,ξ1,ξ′,ξ1′∈X^ and ξ≈ξ′ and ξ1≈ξ1′. By the triangle inequality we see that d^(ξ1,ξ1′)≤d^(ξ1,ξ)+d^(ξ,ξ′)+d^(ξ′,ξ1′) and hence d^(ξ1,ξ1′)≤0+d^(ξ,ξ′)+0. Similarly, d^(ξ,ξ′)≤d^(ξ1,ξ1′) which implies that d^(ξ,ξ′)=d^(ξ1,ξ1′). Hence, d~ is well defined. It is obvious that d~ is quasi-pseudometric. To prove that (X~,d~) is a T0-quasi-pseudometric space suppose that d~(Ξ,Ξ′)=d~(Ξ′,Ξ)=0. Then, d^(ξ,ξ′)=d^(ξ′,ξ)=0. Suppose that ξ,ξ′∈X¯. Let (xn)n∈ℕ∈𝒜ξ and (xm′)m∈ℕ∈𝒜ξ′. Then, by Proposition 18 we conclude that (xn)n∈ℕ and (xm′)m∈ℕ are right d-cofinal. Therefore, 𝒜ξ=𝒜ξ′. If ξ=ϕ(x) and ξ′=ϕ(x′) for some x,x′∈X, then d^(ϕ(x),ϕ(x′))=d^(ϕ(x′),ϕ(x))=0 which implies that d(x,x′)=d(x′,x)=0. Since (X,d) is T0 we conclude that x=x′. Hence, in any case we have Ξ=Ξ′. Consequently, (X~,d~) is a T0-quasi-pseudometric space.Proposition 30.
If(X,d) is a T0-quasi-pseudometric space, then φ=ϕq∘ϕ:(X,d)→(X~,d~) is an isometric embedding.Proof.
We have thatφ=ϕq∘ϕ is an isometry, since both ϕq and ϕ are isometries. On the other hand, for any x,y∈X, ϕ(x)=ϕ(y) implies that d^(ϕ(x),ϕ(y))=0=d^(ϕ(y),ϕ(x)). Then, Proposition 22 implies that d(x,y)=0=d(y,x). Hence, x=y because (X,d) is a T0-quasi-pseudometric space. Therefore, ϕ is injective which implies that φ is an isometric embedding.Theorem 31.
Let(X,d) be a T0-quasi-pseudometric space. Then (X~,d~) is a T0κ-completion of (X,d).Proof.
Let(Ξγ)γ∈ℕ be a κ-Cauchy sequence in (X~,d~). Therefore, since ϕq is a surjective isometry, ϕq-1((Ξγ)γ∈ℕ) is a κ-Cauchy sequence in (X^,d^). Thus for some ξ∈X^, ϕq-1((Ξγ)γ∈ℕ) converges to ξ. It follows that (Ξγ)γ∈ℕ converges to ϕq(ξ)=Ξ since ϕq is an isometry. So (X~,d~) is κ-complete T0-quasi-pseudometric space (see Proposition 29). It remains to prove that the set φ(X)=(ϕq∘ϕ)(X) is dense in (X~,d~). Indeed, suppose that Ξ∈X~. Then, since ϕq is a surjective isometry, there exists ξ∈X^ such that ϕq(ξ)=Ξ. Let (xn)n∈ℕ∈𝒜ξ. Then, by Definition 17 we have that ϕ(xn)→ξ. Therefore, ϕq(ϕ(xn)) converges to ϕq(ξ)=Ξ with respect to τd~. It follows that φ(X) is dense in (X~,d~) which implies that (X~,d~) is a κ-completion of (X,d) (see Definition 16).Definition 32.
Let(X,d) be a T0-quasi-pseudometric space. Then the T0-quasi-pseudometric space (X~,d~) previously defined will be called the standard  κ-completion of (X,d).


## 3. A Categorical View of theκ-Completion

As already pointed out, a natural problem which arises in quasi-pseudometric spaces is to introduce a convenient notion of completeness and to construct a satisfactory theory of completion. What do we mean by a “satisfactory theory” of completeness and completion for quasi-pseudometric spaces? According to Doitchinov, the answer to that question is based on certain restrictions that we must have. These restrictions are separated in two groups as follows.(α)
A notion of Cauchy sequence in any quasi-pseudometric space has to be defined in such a manner that the following requirements are fulfilled:
(i)
every convergent sequence is a Cauchy sequence;(ii)
when(X,d) is a pseudometric space, the Cauchy sequences are the usual ones.(β)
A standard construction of a completion(X^,d^) of any quasi-pseudometric space (X,d) may need the following.
(iii)
If(X,d)⊆(Y,ρ), then (X^,d^)⊆(Y^,ρ^), where inclusions are understood as quasimetric embeddings (an embedding between two objects X and Y is an injective (one-to-one) and structure-preserving map f:X→Y. A structure on a set consists of additional mathematical objects (topological structures, orders, and equivalence relations) that in some manner attach to the set. The precise meaning of “structure-preserving” depends on the kind of mathematical structure of which X and Y are instances. In general topology, an embedding is a homeomorphism or topological isomorphism (a bicontinuous function between two topological spaces) onto its image. More explicitly, a map f:X→Y between topological spaces X and Y is an embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y). In quasimetric spaces such homeomorphisms are called isometries or quasimetric embeddings.) and the second one is an extension of the former.(iv)
In the case when(X,d) is a metric space, (X^,d^) is the usual metric completion of (X,d).After the paper of Künzi and Schellekens [15] another requirement must be added in (β). This is the following.(v)
The constructed completion must be an idempotent operation; that is, theκ-completion of the κ-completion of (X,d) is isomorphic to the κ-completion of (X,d).Theκ-completion differs with all other previously introduced completion theories for quasi-pseudometric spaces. More precisely, in metric spaces, symmetry enables us to construct the classical completion by using Cauchy sequences instead of equivalence classes of Cauchy sequences. Similarly in the κ-procedure, since right (resp. left) d-cofinality between right K-Cauchy sequences (resp. left K-Cauchy cosequences) is an equivalence relation, we use Cauchy sequences instead of κ-cuts. On the other hand, in metric spaces, the notions of Cauchy sequence, right K-Cauchy sequence, and left K-Cauchy cosequence coincide. Then, in the κ-procedure, every class of equivalent Cauchy sequences of the classical case, identifies with the class of right K-Cauchy sequences of a κ-cut and vice versa. We now show that the notion of κ-completeness and the constructed κ-completion satisfy the requirements posed by Doitchinov [1, page 128] as well as the requirement (v) previously posed.The following two propositions show that theκ-completion satisfies the requirements (i) and (ii) previously stated.Proposition 33.
Every convergent rightK-Cauchy sequence is a κ-Cauchy sequence.Proof.
In fact, if a rightK-Cauchy sequence (xn)n∈ℕ converges to a point x of X, then it belongs to the first class of a κ-cut ϕi(x)∈Φ(x). The validity of this proposition follows from Definition 11.Clearly, in the case of metric spaces, the notions of converging rightK-Cauchy sequence and converging sequence coincide.The following proposition is obvious.Proposition 34.
If(X,d) is a pseudometric space, then the notion of κ-Cauchy sequence coincides with the classical one.Remark 35.
Doitchinov in [1, Proposition 1] shows that if a sequence (xn)n∈ℕ converges to a point, say x, then (xn)n∈ℕ has x as its cosequence, so it is a D-Cauchy sequence. (A sequence (xn)n∈ℕ in the quasimetric space (X,d) is called D-Cauchy sequence provided that for any κ∈ℕ there exists a yκ∈X and Nκ∈ℕ such that d(ym,xn)<1/κ when m,n>Nκ.) In another point in [1, page 130], he also says that a “good” D-Cauchy sequence is that which remains D-Cauchy sequence even if we withdraw the limit point from the space. We recall his counterexample as follows: let (X,d) be a quasimetric space, X={xn∣n=0,1,2,…} and
(62)d(xn,xm)={1form>0,m≠n,1nform=0,n≠0,0form=n.
The sequence (xn)n∈ℕ converges to x0, so it must be a D-Cauchy sequence. However, if we withdraw the limit point x0, the space becomes a discrete metric space, so (xn)n∈ℕ ceases to be a D-Cauchy sequence. Of course, this conclusion contradicts with Doitchinov’s first aspect. It is easy to see that with the notion of κ-Cauchy sequence we avoid this undesirable situation. In fact, (xn)n∈ℕ is not a κ-Cauchy sequence.The requirement (iii) of Doitchinov concerns the categorical aspect of a completion. It is well known that the class ofD-complete quasimetric spaces (a quasimetric space is called D-complete if every D-Cauchy sequence is convergent) defines a category 𝒬ℳ𝒮 provided that we assume as morphisms the quasiuniformly continuous functions. Bonsangue et al. [16] defines the category 𝒢ℳ𝒮 with complete generalized metric spaces (a generalized metric space consists of a set X together with a distance function d:X×X→[0,∞], satisfying d(x,x)=0 and d(x,y)<d(x,z)+d(z,y), for all x,z, and y in X. We remark that traditionally a quasi-pseudometric space is required to take finite values. However the Yoneda completion of [16] involves generalized metric which allows for infinite distances) as objects and nonexpansive maps (a function f between two quasi-pseudometric spaces (X,d) and (Y,ρ) is non-expansive provided that ρ(f(x),f(y))≤d(x,y) for all x,y∈X.) as morphisms.We define a new category𝒬𝒫ℳ𝒮 whose objects are the T0 quasi-pseudo-metric spaces and morphisms are the κ-cut preserving functions defined below (Definition 39).Definition 36.
A functionf between two T0 quasi-pseudometric spaces (X,d) and (Y,ρ) is called quasiuniformly continuous if for any ϵ>0 there is a δ>0 such that ρ(x,x′)<ϵ whenever d(x,x′)<δ.Definition 37.
Two sequences(xn)n∈ℕ and (yσ)σ∈ℕ in a T0 quasi-pseudometric space (X,d) form a κ-Cauchy pair of sequences if there exists a κ-cut ξ=(𝒜,ℬ) in X such that (xn)n∈ℕ∈𝒜 and (yσ)σ∈ℕ∈ℬ. Two κ-Cauchy pairs of sequences are κ-compatible if they belong to the same κ-cut.Definition 38.
Let(X,d), (Y,ρ) be T0 quasi-pseudometric spaces and let f:X→Y be a continuous function. Let also ((xn)n∈ℕ, (yσ)σ∈ℕ) be a κ-Cauchy pair of sequences in (X,d). We say that a κ-Cauchy pair ((xm)m∈ℕ, (ys)s∈ℕ) of sequences in (X,d) is (κ,f)-compatible to ((xn)n∈ℕ, (yσ)σ∈ℕ) if it satisfies the following conditions:(i)
((xn)n∈ℕ,(yσ)σ∈ℕ) and ((xm)m∈ℕ,(ys)s∈ℕ) are κ-compatible;(ii)
((f(xm))m∈ℕ, (f(ys))s∈ℕ) is a κ-Cauchy pair of sequences in Y.Definition 39.
Let(X,d), (Y,ρ) be T0 quasi-pseudometric spaces. A continuous function f:X→Y is called κ-cut preserving if it satisfies the following conditions.(i)
For eachκ-Cauchy pair ((xn)n∈ℕ,(yσ)σ∈ℕ) of sequences in X, there exists a κ-Cauchy pair ((xm)m∈ℕ,(ys)s∈ℕ) of sequences in X such that ((xm)m∈ℕ,(ys)s∈ℕ) is (κ,f)-compatible to ((xn)n∈ℕ,(yσ)σ∈ℕ).(ii)
If((xn)n∈ℕ,(yσ)σ∈ℕ),((xn′)n∈ℕ,(yσ′)σ∈ℕ), ((xm)m∈ℕ,(ys)s∈ℕ) and ((xm′)m∈ℕ,(ys′)s∈ℕ) are κ-Cauchy pairs of sequences such that ((xm)m∈ℕ,(ys)s∈ℕ) is (κ,f)-compatible to ((xn)n∈ℕ,(yσ)σ∈ℕ) and ((xm′)m∈ℕ,(ys′)s∈ℕ) is (κ,f)-compatible to ((xn′)n∈ℕ,(yσ′)σ∈ℕ), then, the κ-compatibility (resp. non-κ-compatibility) of ((xn)n∈ℕ,(yσ)σ∈ℕ) and ((xn′)n∈ℕ,(yσ′)σ∈ℕ) implies the κ-compatibility (non-κ-compatibility) of ((f(xm))m∈ℕ,(f(ys))s∈ℕ) and ((f(xm′))m∈ℕ,(f(ys′))s∈ℕ), respectively.By the previously mentioned definition, it is clear that a cut preserving functionf takes distinct elements of X to distinct elements of Y. On the other hand, a cut preserving cut f is one that “embeds” X as a subset of Y. In view of the Definitions 8, 11, and 37, Proposition 22, and Theorem 27, one obtains that the quasi-pseudometric embedding ϕ is a κ-cut preserving function.Doitchinov [1] uses quasiuniformly continuous functions defined in quasimetric spaces. Lemma  7 in [1] shows that these functions are κ-cut preserving functions.Similarly to what is in the theory of metric spaces, the categorical notion of aT0 quasi-pseudometric completion is the following: a complete T0 quasi-pseudometric space (X~,d~) is called a κ-completion of a given T0 quasi-pseudometric space (X,d) if there exists a κ-cut preserving function φ:(X,d)→(X~,d~) such that(a)
φ(X) is dense in (X~,d~);(b)
the quasi-pseudometric(X,d) is the inverse image of (X~,d~) under φ;(c)
for anyκ-cut preserving function f:(X,d)→(Y,ρ), where (Y,ρ) is a κ-complete T0 quasi-pseudometric space, there is a κ-cut preserving function f*:(X~,d~)→(Y,ρ) with f=f*∘φ.The mappingφ is called the canonical quasi-pseudometric embedding of (X,d) into (X~,d~).Proposition 40.
Let(X,d), (Y,ρ) be two T0 quasi-pseudometric spaces, let (Y,ρ) be κ-complete, and let f:(X,d)→(Y,ρ) be a κ-cut preserving function. Then, there exists a κ-cut preserving function f*:(X~,d~)→(Y,ρ) such that f=f*∘φ.Proof.
Let(X,d), (Y,ρ), and f be as in the hypothesis of the proposition. Now define a mapping f*:(X~,d~)→(Y,ρ) by letting f*(Ξ)=limnf(xn) whenever ξ is a κ-cut in X such that ϕq(ξ)=Ξ and (xn)n∈ℕ∈𝒜ξ. Definitions 28 and 39 imply that f* takes distinct elements of X~ to distinct elements of Y. Let Ξ=φ(x) where φ=ϕq∘ϕ:(X,d)→(X~,d~) be the isometric embedding defined in Proposition 30. Then, by the continuity of f at x, we have f*(φ(x))=f(x). It remains to prove that f* is κ-cut preserving. Let ((Ξγ)γ∈ℕ,(Hβ)β∈ℕ) be a pair Cauchy of sequences in X~. Suppose that for each γ,β∈ℕ we have Ξγ=ϕq(ξγ) and Hβ=ϕq(ηβ). Then, let ((ξγ)γ∈ℕ,(ηβ)β∈ℕ) be a pair Cauchy of sequences in X^. Then, by Propositions 25 and 26 and Theorem 27 there is a pair Cauchy ((xn)n∈ℕ, (yσ)σ∈ℕ) of sequences in X such that ((ξγ)γ∈ℕ,(ηβ)β∈ℕ) and ((ϕ(xn))n∈ℕ, (ϕ(yσ))σ∈ℕ) are κ-compatible, (ξ)γ∈ℕ and ((ϕ(xn))n∈ℕ are right d^-cofinal, and (ηβ)β∈ℕ and ((ϕ(yσ))σ∈ℕ are left d^-cofinal. Since f is cut preserving, there exists a Cauchy pair ((xm)m∈ℕ, (ys)s∈ℕ) of sequences satisfying the conditions (i) and (ii) of Definition 39. Since φ=ϕq∘ϕ is an isometry and f=f*∘φ it follows that f* satisfies the conditions (i) and (ii) of Definition 39. Hence, f* is κ-cut preserving.The previous proposition shows that the standardκ-completion is the smallest κ-complete quasi-pseudometric space containing X in the sense that any other κ-complete space Y containing X as a subspace also contains X as a subspace.The next two propositions show that theκ-completion satisfies the requirements (iii) and (iv) posed by Doitchinov.Proposition 41.
Let(X,d), (Y,ρ) be two T0 quasi-pseudometric spaces, and let (X~,d~), (Y~,ρ~) be their standard κ-completions with canonical maps φX:(X,d)→(X~,d~) and φY:(Y,ρ)→(Y~,ρ~). If f:(X,d)→(Y,ρ) is a κ-cut preserving function, then there exists a κ-cut preserving function f*:(X~,d~)→(Y~,ρ~).Proof.
The existence of aκ-cut preserving function f*:(X~,d~)→(Y~,ρ~) follows from Proposition 40 when one applies it to the function φY∘f.Proposition 42.
If(X,d) is a metric space, then the standard κ-completion (X~,d~) coincides with the usual metric completion.Proof.
Let(X,d) be a metric space. Then in the κ-procedure the notions of right K-Cauchy sequence, left K-Cauchy cosequence, and Cauchy sequence coincide. Hence, every class of equivalent Cauchy sequences of the classical case identifies with a κ-cut and vice versa. Hence, the new elements of the two completions, the first of the classical completion and the second of the κ-completion, coincide. The rest is obvious.Finally, let us discuss the requirement (v) previously stated in the introduction of this section. It is well known in classical metric space theory that Cauchy completion is idempotent. One can also verify that the bicompletion as well as the Doitchinov completion is an idempotent operation. The idempotency of the well known completions for quasi-pseudometric spaces has not been studied. There are also two main types of domain theoretic completions, the Smyth completion [17] and the Yoneda completion [16]. The Smyth completion is idempotent, while the Yoneda completion is not (see [15, page 164]). Recently, Künzi and Schellekens in [15] introduce a general net-version for the Yoneda completion and characterize the family of generalized metric spaces (generalized metric spaces are a common generalization of preorders and ordinary metric spaces. Such spaces consist of a set X together with a distance function d:X×X→[0,∞] satisfying d(x,x)=0 and d(x,z)≤d(x,y)+d(y,z), for all x,y and z in X.) for which the Yoneda completion is idempotent.We recall the classical definition of idempotency (adapted to the specific case of theκ-completion). In the following (X~~,d~~) denotes the κ-completion of the κ-completion (X~,d~).Definition 43.
Theκ-completion of a T0 quasi-pseudometric space (X,d) is idempotent if and only if there exists an isometry f:(X~,d~)→(X~~,d~~) such that for each Ξ~∈X~~, Ξ~=f(Ξ) for some Ξ∈(X~,d~).Proposition 44.
Theκ-completion is idempotent.Proof.
Let(X,d) be a quasi-pseudometric space and let (X~,d~) be its κ-completion. Let us denote by X~~ the set of κ-cuts in X~ and let Ξ~=(𝒜Ξ~,ℬΞ~)∈X~~. Suppose that 𝒜Ξ~={(Ξγ(i)i)γ(i)∈ℕ∣i∈I} and ℬξ~={(Hδ(j)j)δ(j)∈ℕ∣j∈J}. Then, the families 𝒜ξ^={(ξγ(i)i)γ(i)∈ℕ∣i∈I} and ℬξ^={(ηδ(j)j)δ(j)∈ℕ∣j∈J}, where ϕq(ξγ(i)i)=Ξγ(i)i and ϕq(ηδ(j)j)=Hδ(j)j, define a κ-cut ξ^=(𝒜ξ^,ℬξ^) in X^. Then, according to Theorem 27, to each κ-cut ξ^ there corresponds a κ-cut ξ∈(X,d). Let Ξ=ϕq(ξ) and let φ~ be the quasi-pseudometric embedding of (X~,d~) into (X~~,d~~) as it is defined in Definition 9. Then, Ξ~=φ~(Ξ). Therefore, we have X~~=φ~(X~). Next, for each Ξ~′,Ξ~′′∈X~~ we have
(63)d~~(Ξ~′,Ξ~′′)=d~~(φ~(Ξ′),φ~(Ξ′′))=d~(Ξ′,Ξ′′)=d^(ξ′,ξ′′),
where Ξ′=ϕq(ξ′) and Ξ′′=ϕq(ξ′′). Clearly, φ~ is an isometry. To complete the proof we must show that (X~~,d~~) is κ-complete. Indeed, let (Ξ~γ)γ∈ℕ be a κ-Cauchy sequence in (X~~,d~~). Then, there exists a κ-cut Ξ~~=(𝒜Ξ~~,ℬΞ~~) in (X~~,d~~) such that (Ξ~γ)γ∈ℕ∈𝒜Ξ~~. Suppose that 𝒜Ξ~~={(Ξ~γ(i)i)γ(i)∈ℕ∣i∈I}, ℬΞ~~={(H~δ(j)j)δ(j)∈ℕ∣j∈J} and (Ξ~γ)γ∈ℕ=(Ξ~γ(i0)i0)γ(i0)∈ℕ for some i0∈I. For each i∈I and each j∈J we have Ξ~γ(i)i=φ~(Ξγ(i)i) and H~δ(j)i=φ~(Hδ(j)j). On the other hand, Ξγ(i)i=ϕq(ξγ(i)i) and Hδ(j)j=ϕq(ηδ(j)j) for some ξγ(i)i,ηδ(j)j∈(X,d). Next, we define ξ^=(𝒜ξ^,ℬξ^), where 𝒜ξ^={(ξγ(i)i)γ(i)∈ℕ∣i∈I} and ℬξ^={(ηδ(j)i)δ(j)∈ℕ∣j∈J}. By Theorem 27, there exists a κ-cut ξ∈(X^,d^) such that, for each γ(i)∈ℕ, (ξγ(i)i)γ(i)∈ℕ→ξ with respect to τ(d^). By definition of d~, it follows that (Ξγ(i)i)γ(i)∈ℕ→Ξ with respect to τ(d~), where Ξ=ϕq(ξ). Since φ~ is an isometry, Proposition 25 implies that (Ξ~γ(i)i)γ(i)∈ℕ and (φ~(Ξγ(i)i))γ(i)∈ℕ are right d~~-cofinal. Therefore, Corollary 7 implies that (Ξ~γ(i)i)γ(i)∈ℕ converges to φ~(Ξ) with respect to τ(d~~). It follows that (Ξ~γ)γ∈ℕ=(Ξ~γ(i0)i0)γ(i0)∈ℕ converges to φ~(Ξ) with respect to τ(d~~).



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