A Solution to the Completion Problem for Quasi-Pseudometric Spaces

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 given 𝑇₀ 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.

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 𝑑_𝑠 we denote the (pseudo)metric given by 𝑑_𝑠=max{𝑑(𝑥,𝑦),𝑑⁻¹(𝑥,𝑦)}. Each quasi-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>0}, where 𝐵_𝑑(𝑥,𝑟)={𝑦∈𝑋∶𝑑(𝑥,𝑦)<𝑟}. A quasi-pseudometric space is 𝑇₀ if its associated topology 𝜏_𝑑 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 𝜏_𝑑, 𝜏_𝑑⁻¹, and 𝜏_𝑑𝑠, 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 (𝑥_𝑛)_𝑛∈ℕ in a quasi-pseudometric space (𝑋,𝑑) is called Cauchy sequence, if for each 𝜎∈ℕ there are a 𝑦_𝜎∈𝑋 and a 𝑁_𝜎∈ℕ such that 𝑑(𝑦_𝜎,𝑥_𝑛)<1/𝜎 when 𝑛>𝑁_𝜎. 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 (𝑥_𝑛)_𝑛∈ℕ and (𝑥_𝑚)_𝑚∈ℕ are two sequence in (𝑋,𝑑) and 𝑥,𝑦∈𝑋, then from 𝑑(𝑥,𝑥_𝑛)≤𝑟₁ for each 𝑛, 𝑑(𝑥_𝑚,𝑦)≤𝑟₂ for each 𝑚, and lim_𝑚,𝑛𝑑(𝑥_𝑚,𝑥_𝑛)=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^∗,𝑑^∗_𝑠) is (isometric to) the completion of the metric space (𝑋,𝑑_𝑠). We recall that (𝑋,𝑑) is called bicomplete provided that each 𝑑_𝑠-Cauchy filter converges in (𝑋,𝑑_𝑠). Let 𝐴,𝐵 be nonempty subsets of 𝑋. The 2-diameter from 𝐴 to 𝐵 is defined by Φ_𝑑(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 inf_{𝐹∈ℱ,𝐺∈𝒢}Φ_𝑑(𝐹,𝐺)=0. An arbitrary Cauchy filter pair (ℱ,𝒢) on 𝑋 is said to converge to 𝑥∈𝑋 provided that inf_𝐺∈𝒢Φ_𝑑(𝑥,𝐺)=0 and inf_𝐹∈ℱΦ_𝑑(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)≤inf_𝐺∈𝒢Φ_𝑑(𝑥,𝐺)+inf_𝐹∈ℱΦ_𝑑(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^′))=inf_{𝐹∈ℱ,𝐺′∈𝒢}Φ_𝑑(𝐹,𝐺^′)=inf_{𝐹∈ℱ,𝐺′∈𝒢}sup_{𝑓∈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⁺𝑋(𝑞₊(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 (𝑋^𝛽,𝑑^𝛽) 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 (𝑋^𝛽,𝑑^𝛽) is 𝐵-complete. The space (𝑋^𝛽,𝑑^𝛽) will be called the standard  𝐵-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]). Let 𝑋={−1/(𝑛+1),1/(𝑛+1),𝑛∈ℕ}. For each 𝑥,𝑦∈𝑋, let 𝑑𝑥,𝑦=1if𝑥<0<𝑦,0if𝑦≤𝑥,min1,∣𝑥−𝑦∣otherwise. (1) 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 1=𝑑−14,14≰inf_𝐺∈𝒢Φ_𝑑−14,𝐺+inf_𝐹∈ℱΦ_𝑑1𝐹,4=14+14=12. (2)
Therefore, the 𝐵-completion (𝑋^𝛽,𝑑^𝛽) 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, called 𝜅-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 (𝑦_𝜎)_𝜎∈ℕ is called a cosequence to (𝑥_𝑛)_𝑛∈ℕ, if for any 𝜀>0 there are 𝜎_𝜀,𝑛_𝜀∈ℕ such that 𝑑(𝑦_𝜎,𝑥_𝑛)<𝜀 when 𝜎≥𝜎_𝜀 and 𝑛≥𝑛_𝜀. In this case, we write 𝑑(𝑦_𝜎,𝑥_𝑛)→0 or lim_𝜎,𝑛𝑑(𝑦_𝜎,𝑥_𝑛)=0. Generally, given two sequences (𝑥_𝑛)_𝑛∈ℕ and (𝑥_𝑚)_𝑚∈ℕ in (𝑋,𝑑), we write lim_𝑛,𝑚𝑑(𝑥_𝑛,𝑥_𝑚)=𝑟 if for any 𝜀>0 there are 𝑛_𝜀,𝑚_𝜀∈ℕ such that |𝑑(𝑥_𝑛,𝑥_𝑚)−𝑟|<𝜀 when 𝑛≥𝑛_𝜀 and 𝑚≥𝑚_𝜀.

Definition 2 (see [3]). A sequence (𝑥_𝑛)_𝑛∈ℕ of a quasi-pseudometric space (𝑋,𝑑) is called right (resp. left) 𝐾- Cauchy, if for any 𝜀>0 there is an 𝑛_𝜀∈ℕ such that 𝑑(𝑥_𝑛,𝑥_𝑛′)<𝜀 for each 𝑛≥𝑛^′≥𝑛_𝜀 (resp. 𝑛^′≥𝑛≥𝑛_𝜀). One calls 𝑛_𝜀 (resp. 𝑥_𝑛𝜀) the extreme index (resp. extreme point) of (𝑥_𝑛)_𝑛∈ℕ for 𝜀.

Definition 3. Let (𝑋,𝑑) be a quasi-pseudometric space and let (𝑥_𝑛)_𝑛∈ℕ, (𝑥_𝑚)_𝑚∈ℕ be two sequences on it. One says that (𝑥_𝑛)_𝑛∈ℕ is right (resp. left) 𝑑-cofinal to  (𝑥_𝑚)_𝑚∈ℕ, if for each 𝜀>0 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.

Proposition 4. Let (𝑥_𝑛)_𝑛∈ℕ be a right 𝐾-Cauchy (left 𝐾-Cauchy) sequence in a quasi-pseudometric space (𝑋,𝑑) with a subsequence (𝑥_𝑛𝜁)_{𝑛𝜁∈ℕ}. Then, (𝑥_𝑛)_𝑛∈ℕ and (𝑥_𝑛𝜁)_{𝑛𝜁∈ℕ} are right (resp. left) 𝑑-cofinal.

Proof. Let (𝑥_𝑛)_𝑛∈ℕ and (𝑥_𝑛𝜁)_{𝑛𝜁∈ℕ} be as in the statement of the proposition. Let 𝜀>0 be given. We know that from some index 𝑛_𝜀 onwards we have that 𝑛≥𝑛^′≥𝑛_𝜀 implies that 𝑑(𝑥_𝑛,𝑥_𝑛′)<𝜀/2. Now if 𝑛≥𝑛_𝜀 we can find 𝑁_𝜁∈ℕ such that 𝑁_𝜁>𝑛 (as the 𝑛_𝜁 form a strictly increasing sequence of integers). Hence, for each 𝑛_𝜁≥𝑁_𝜁 we have 𝑑(𝑥_𝑛𝜁,𝑥_𝑛)<𝑑(𝑥_𝑛𝜁,𝑥_𝑁𝜁)+𝑑(𝑥_𝑁𝜁,𝑥_𝑛)<𝜀/2+𝜀/2=𝜀. Hence, (𝑥_𝑛)_𝑛∈ℕ is right 𝑑-cofinal to (𝑥_𝑛𝜁)_{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 (𝑥_𝑛)_𝑛∈ℕ and (𝑥_𝑚)_𝑚∈ℕ be two right 𝑑-cofinal sequences. Suppose that (𝑦_𝜎)_𝜎∈ℕ is a cosequence of (𝑥_𝑚)_𝑚∈ℕ. Fix an 𝜀>0. Then there exists 𝜎_𝜀, 𝑚_𝜀∈ℕ such that 𝑑(𝑦_𝜎,𝑥_𝑚)<𝜀/2 for 𝜎≥𝜎_𝜀 and 𝑚≥𝑚_𝜀. On the other hand, there is 𝑛_𝜀∈ℕ with the following property: for each 𝑛≥𝑛_𝜀 there exists 𝑚_𝑛∈ℕ such that 𝑑(𝑥_𝑚,𝑥_𝑛)<𝜀/2 whenever 𝑚≥𝑚_𝑛. If 𝑚(𝑛)=max{𝑚_𝜀,𝑚_𝑛}, then 𝑑(𝑦_𝜎,𝑥_𝑛)≤𝑑(𝑦_𝜎,𝑥_𝑚(𝑛))+𝑑(𝑥_𝑚(𝑛),𝑥_𝑛)<𝜀 for each 𝜎≥𝜎_𝜀, 𝑛≥𝑛_𝜀. Hence, (𝑦_𝜎)_𝜎∈ℕ is a cosequence of (𝑥_𝑛)_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, for 𝜀^′≤𝜀, it is 𝑛_𝜀′≥𝑛_𝜀.

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 (𝑥_𝑛)_𝑛∈ℕ∈𝒜 and any (𝑦_𝜎)_𝜎∈ℕ∈ℬ there holds lim_𝜎,𝑛𝑑𝑦_𝜎,𝑥_𝑛=0. (3)(ii)Any two members of the family 𝒜 (resp. ℬ) are right (resp. left) 𝑑-cofinal.(iii)The classes 𝒜 and ℬ are maximal with respect to set inclusion.

We call 𝒜 (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 every 𝑥∈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 𝜏_𝑑 and ℬ consists of left 𝐾-Cauchy cosequences which converges to 𝑥 with respect to 𝜏_𝑑⁻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 𝜏_𝑑 and 𝜏_𝑑⁻¹, respectively, then 𝜙(𝑥)=((𝑥),(𝑥)).

As we can map 𝑋 to the set of all 𝜅-cuts, it is possible that for some 𝑥∈𝑋 there correspond many 𝜅-cuts; let 𝜙_𝑖(𝑥), 𝑖∈𝐼_𝑥 (the index set 𝐼_𝑥 depends on the choice of 𝑥). Then, we define Φ(𝑥)=𝜙_𝑖(𝑥)∣𝑖∈𝐼_𝑥. (4) Without loss of generality, we assume that 𝜙(𝑥)=𝜙_𝑖0(𝑥) for some 𝑖₀∈𝐼_𝑥.

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)∈𝒜_𝜙(𝑥)∩ℬ_𝜙(𝑥)∩𝒜_𝜙(𝑦)∩ℬ_𝜙(𝑦). Thus, 𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥)=0 which implies that 𝑥=𝑦.

Definition 11. One calls 𝜅-Cauchy sequence any right 𝐾-Cauchy sequence which is member of the first class of a 𝜅-cut.

Definition 12. Two right 𝐾-Cauchy sequences (𝑥_𝑛)_𝑛∈ℕ and (𝑥_𝑚)_𝑚∈ℕ defined in a quasi-pseudometric space (𝑋,𝑑) are called 𝜅-equivalent if every left 𝐾-Cauchy cosequence to (𝑥_𝑛)_𝑛∈ℕ is a left 𝐾-Cauchy cosequence to (𝑥_𝑚)_𝑚∈ℕ and vice versa.

Clearly, 𝜅-equivalence defines an equivalence relation on 𝑋.

An immediate consequence of Definitions 8, 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 Corollary 13, 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. A 𝜅-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 a 𝜅-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 𝑋=𝜙(𝑋)𝑋. (5)

Throughout the paper, for every 𝜉∈𝑋, 𝒜_𝜉, ℬ_𝜉 denote the two classes of 𝜉. In this case, we write 𝜉=(𝒜_𝜉,ℬ_𝜉).

Definition 17. Let (𝑋,𝑑) be a quasi-pseudometric space. Suppose that 𝑟 is a nonnegative real number, 𝜉^′,𝜉^′′∈𝑋, (𝑥^′_𝑛)_𝑛∈ℕ∈𝒜_𝜉′, and (𝑥^′′_𝑚)_𝑚∈ℕ∈𝒜_𝜉′′. We put 𝑑(𝜉^′,𝜉^′′)≤𝑟 if(i)𝒜_𝜉′=𝒜_𝜉′′ or(ii)for each 𝜀>0 there are 𝑛^′_𝜀, 𝑚^′′_𝜀∈ℕ such that 𝑑𝑥^′_𝑛,𝑥^′′_𝑚<𝑟+𝜀 (6) when 𝑛≥𝑛^′_𝜀 and 𝑚≥𝑚^′′_𝜀. If 𝜉^′=𝜙(𝑥) for some 𝑥∈𝑋, then the arbitrary sequence (𝑥^′_𝑛)_𝑛∈ℕ always coincides with the fixed sequence, for which 𝑥^′_𝑛=𝑥 for all 𝑛∈ℕ. That is, 𝑑(𝜙(𝑥),𝜉^′′)≤𝑟 if 𝑑𝑥,𝑥^′′_𝑚<𝑟+𝜀 (7) when 𝑚≥𝑚^′′_𝜀.

Then we let 𝑑𝜉^′,𝜉^′′=inf𝑟∣𝑑𝜉^′,𝜉^′′≤𝑟. (8)

Proposition 18. Let 𝜉^′,𝜉^′′∈𝑋, (𝑥^′_𝑛)_𝑛∈ℕ∈𝒜_𝜉′ and (𝑥^′′_𝑚)_𝑚∈ℕ∈𝒜_𝜉′′. Suppose that 𝑑(𝜉^′,𝜉^′′)=0. Then, (𝑥^′′_𝑚)_𝑚∈ℕ is right 𝑑-cofinal to (𝑥^′_𝑛)_𝑛∈ℕ.

Proof. By the suppositions, for each 𝜀>0 there are 𝑛^′_𝜀, 𝑚^′′_𝜀∈ℕ such that 𝑑(𝑥^′_𝑛,𝑥^′′_𝑚)<𝜀 whenever 𝑛≥𝑛^′_𝜀 and 𝑚≥𝑚^′′_𝜀. The rest is evident by Definition 3.

Proposition 19. The truth of 𝑑(𝜉^′,𝜉^′′)≤𝑟 in Definition 17 depends only on 𝜉^′, 𝜉^′′, and 𝑟; it does not depend on the choice of the sequences (𝑥^′_𝑛)_𝑛∈ℕ and (𝑥^′′_𝑚)_𝑚∈ℕ.

Proof. Let 𝜉^′,𝜉^′′, (𝑥^′_𝑛)_𝑛∈ℕ, (𝑥^′′_𝑚)_𝑚∈ℕ, and 𝑟 be as in the supposition of Definition 17. Further, let 𝑑(𝜉^′,𝜉^′′)≤𝑟. Then, for each 𝜀>0 there are 𝑛^′_𝜀, 𝑚^′′_𝜀∈ℕ such that 𝑑𝑥^′_𝑛,𝑥^′′_𝑚𝜀<𝑟+3 (9) when 𝑛≥𝑛^′_𝜀 and 𝑚≥𝑚^′′_𝜀.
We choose two arbitrary sequences (𝑥^′_𝑠)_𝑠∈ℕ∈𝒜_𝜉′ and (𝑥^′′_𝑡)_𝑡∈ℕ∈𝒜_𝜉′′. Since (𝑥^′_𝑛)_𝑛∈ℕ is right 𝑑-cofinal to (𝑥^′_𝑠)_𝑡∈ℕ, there is 𝑁^′_𝜀∈ℕ satisfying the following property: for each 𝑛≥𝑁^′_𝜀 there exists 𝑠_𝑛∈ℕ such that 𝑑𝑥^′_𝑠,𝑥^′_𝑛<𝜀3 (10) whenever 𝑠≥𝑠_𝑛. Let 𝑛^′_𝜀=max{𝑛^′_𝜀,𝑁^′_𝜀}. Then, we have𝑑𝑥^′_𝑠,𝑥^′′_𝑚≤𝑑𝑥^′_𝑠𝑛,𝑥_′𝜀𝑥𝑛+𝑑_′𝜀,𝑥^′′_𝑚𝜀<𝑟+3+𝜀3=𝑟+2𝜀3 (11) whenever 𝑛𝑠≥𝑠_′𝜀 and 𝑚≥𝑚^′′_𝜀.
Analogously, since (𝑥^′′_𝑡)_𝑡∈ℕ is right 𝑑-cofinal to (𝑥^′′_𝑚)_𝑚∈ℕ, there is 𝑡_𝜀∈ℕ satisfying the following property: for each 𝑡≥𝑡_𝜀 there exists 𝑚_𝑡∈ℕ such that 𝑑𝑥^′′_𝑚,𝑥^′′_𝑡<𝜀3 (12) whenever 𝑚≥𝑚_𝑡. Let 𝑚(𝑡,𝜀)=max{𝑚^′′_𝜀,𝑚_𝑡}. Then, by (11) we have 𝑑𝑥^′_𝑠,𝑥^′′_𝑡≤𝑑𝑥^′_𝑠,𝑥^′′_{𝑚(𝑡,𝜀)}+𝑑𝑥^′′_{𝑚(𝑡,𝜀)},𝑥^′′_𝑡<𝑟+2𝜀3+𝜀3=𝑟+𝜀 (13) when 𝑛𝑠≥𝑠_′𝜀 and 𝑡≥𝑡_𝜀.

Proposition 20. Let 𝜉^′,𝜉^′′∈𝑋, (𝑥^′_𝑛)_𝑛∈ℕ∈𝒜_𝜉′, and (𝑥^′′_𝑚)_𝑚∈ℕ∈𝒜_𝜉′′. Then, 𝑑𝜉^′,𝜉^′′=lim_𝑛,𝑚𝑑𝑥^′_𝑛,𝑥^′′_𝑚. (14)

Proof. Let 𝑑(𝜉^′,𝜉^′′)=𝑟. Then, for any 𝜀>0 there are 𝑛^′_𝜀, 𝑚^′′_𝜀∈ℕ such that 𝑑𝑥^′_𝑛,𝑥^′′_𝑚<𝑟+𝜀 (15) when 𝑛≥𝑛^′_𝜀 and 𝑚≥𝑚^′′_𝜀. To prove that 𝑟−𝜀<𝑑(𝑥^′_𝑛,𝑥^′′_𝑚) for 𝑛≥𝑛^′_𝜀 and 𝑚≥𝑚^′′_𝜀, suppose to the contrary that there exist a subsequence (𝑥^′_𝑛𝜁)_{𝑛𝜁∈ℕ} of (𝑥^′_𝑛)_𝑛∈ℕ and a subsequence (𝑥^′′_𝑚𝜐)_{𝑚𝜐∈ℕ} of (𝑥^′′_𝑚)_𝑚∈ℕ such that for all 𝑥^′_𝑛𝜁,𝑥^′′_𝑚𝜐 there holds 𝑑(𝑥^′_𝑛𝜁,𝑥^′′_𝑚𝜐)≤𝑟−𝜀. Then, since Propositions 4, 5, and 6 imply that (𝑥^′_𝑛𝜁)_{𝑛𝜁∈ℕ}∈𝒜_𝜉′ and (𝑥^′′_𝑚𝜐)_{𝑚𝜐∈ℕ}∈𝒜_𝜉′′, it follows, according to Definition 17, that 𝑑(𝜉^′,𝜉^′′)≤𝑟−𝜀, a contradiction. Therefore, we have lim_𝑛,𝑚𝑑(𝑥^′_𝑛,𝑥^′′_𝑚)=𝑟.

Proposition 21. 𝑑 is a quasi-pseudometric.

Proof. From Definition 17 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)𝒜_𝜉≠𝒜_𝜉′ and 𝒜_𝜉′≠𝒜_𝜉′′. Suppose that 𝑑(𝜉,𝜉^′)=𝑟₁, 𝑑(𝜉^′,𝜉^′′)=𝑟₂, (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉, (𝑥^′_𝑚)_𝑚∈ℕ∈𝒜_𝜉′, and (𝑥^′′_𝑠)_𝑠∈ℕ∈𝒜_𝜉′′. Then, for any 𝜀>0 there are 𝑛_𝜀, 𝑚^′_𝜀>0 such that 𝑑(𝑥_𝑛,𝑥^′_𝑚)<𝑟₁+𝜀/2 whenever 𝑛≥𝑛_𝜀 and 𝑚≥𝑚^′_𝜀. Similarly, there are 𝑚^′_𝜀, 𝑠^′′_𝜀>0 such that 𝑑(𝑥^′_𝑚,𝑥^′′_𝑠)<𝑟₂+𝜀/2 whenever 𝑚≥𝑚^′_𝜀 and 𝑠≥𝑠^′′_𝜀. Let 𝑀₀=max{𝑚_𝜀,𝑚^′_𝜀}. Then, 𝑑(𝑥_𝑛,𝑥^′′_𝑠)≤𝑑(𝑥_𝑛,𝑥^′_𝑀0)+𝑑(𝑥^′_𝑀0,𝑥^′′_𝑠)<𝑟₁+𝑟₂+𝜀 for each 𝑛≥𝑛_𝜀, 𝑠≥𝑠^′′_𝜀. Hence, according to Definition 17, we have 𝑑𝜉,𝜉^′′≤𝑟₁+𝑟₂=𝑑𝜉,𝜉^′+𝑑𝜉^′,𝜉^′′. (16)(ii)𝒜_𝜉≠𝒜_𝜉′ and 𝒜_𝜉′=𝒜_𝜉′′. Suppose that 𝑑(𝜉,𝜉^′)=𝑟. Since 𝒜_𝜉′=𝒜_𝜉′′, Definition 17 implies that 𝑑(𝜉,𝜉^′)≤𝑟 and 𝑑(𝜉^′,𝜉^′′)=0. Hence 𝑑𝜉,𝜉^′′≤𝑟=𝑑𝜉,𝜉^′+𝑑𝜉^′,𝜉^′′. (17)(iii)𝒜_𝜉=𝒜_𝜉′ and 𝒜_𝜉′≠𝒜_𝜉′′. This case is proved similar to the previous one.(iv)𝒜_𝜉=𝒜_𝜉′ and 𝒜_𝜉′=𝒜_𝜉′′. This case is trivial.

After the definition of the quasi-pseudometric 𝑑, 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 any 𝑥,𝑦∈𝑋 there holds 𝑑(𝜙(𝑥),𝜙(𝑦))=𝑑(𝑥,𝑦).

Proof. The assertion of the proposition follows from the fact that 𝑥∈𝒜_𝜙(𝑥), 𝑦∈𝒜_𝜙(𝑦), and Proposition 20.

Proposition 23. For any 𝜉∈𝑋,(a)if (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉 then 𝑑(𝜉,𝜙(𝑥_𝑛))→0;(b)if (𝑦_𝜎)_𝜎∈ℕ∈ℬ_𝜉 then 𝑑(𝜙(𝑦_𝜎),𝜉)→0.

Proof. (a)  Let 𝜀>0. Then, there exists 𝑛_𝜀>0 such that 𝑑(𝑥_𝑛,𝑥_𝑛′)<𝜀/2 whenever 𝑛≥𝑛^′≥𝑛_𝜀. Now pick a right 𝐾-Cauchy sequence (𝑥_𝑚)_𝑚∈𝑁 of 𝒜_𝜉. Since (𝑥_𝑛)_𝑛∈ℕ is right 𝑑-cofinal to (𝑥_𝑚)_𝑚∈ℕ, there is 𝑛^∗_𝜀∈ℕ satisfying the following property: for each 𝑛≥𝑛^∗_𝜀 there exists 𝑚_𝑛∈ℕ such that 𝑑𝑥_𝑚,𝑥_𝑛<𝜀2 (18) whenever 𝑚≥𝑚_𝑛. Suppose that 𝑛_𝜀=max{𝑛_𝜀,𝑛^∗_𝜀}. Fix an 𝑛≥𝑛_𝜀 and let (𝑥_𝑠)_𝑠∈ℕ∈𝒜_{𝜙(𝑥𝑛)}. Then, 𝑥_𝑠→𝑥_𝑛 and so there exists an 𝑠_𝜀∈ℕ such that 𝑑(𝑥_𝑛,𝑥_𝑠)<𝜀/2 for each 𝑠≥𝑠_𝜀. Hence, 𝑑𝑥_𝑚,𝑥_𝑠≤𝑑𝑥_𝑚,𝑥_𝑛+𝑑𝑥_𝑛,𝑥_𝑠<𝜀2+𝜀2=𝜀 (19) for 𝑚≥𝑚_𝑛 and 𝑠≥𝑠_𝜀. Since (𝑥_𝑚)_𝑚∈ℕ∈𝒜_𝜉 and (𝑥_𝑠)_𝑠∈ℕ is an arbitrary right 𝐾-Cauchy sequence of 𝒜_{𝜙(𝑥𝑛)}, we conclude that 𝑑(𝜉,𝜙(𝑥_𝑛))<𝜀. Hence, 𝑑(𝜉,𝜙(𝑥_𝑛))<𝜀 whenever 𝑛≥𝑛_𝜀.
(b) The proof is similar to that of (a).

Proposition 23 implies the following corollary.

Corollary 24. The set 𝜙(𝑋) is dense in (𝑋,𝑑).

Proposition 25. Let (𝜉_𝛾)_𝛾∈ℕ be a nonconstant right 𝐾-Cauchy sequence of (𝑋,𝑑) without last element. Then there exists a right 𝐾-Cauchy sequence (𝑥_𝑛)_𝑛∈ℕ of (𝑋,𝑑) such that the sequences (𝜉_𝛾)_𝛾∈ℕ and (𝜙(𝑥_𝑛))_𝑛∈ℕ are right 𝑑-cofinal right 𝐾-Cauchy sequences.

Proof.
(a)  The Construction of the Right  𝐾-Cauchy Sequence  (𝑥_𝑛)_𝑛∈ℕ  of  𝑋. Let (𝜉_𝛾)_𝛾∈ℕ be a 𝜅-Cauchy sequence in 𝑋. Without loss of generality we can assume that, for each 𝛾∈ℕ, 𝒜_𝛾≠𝒜_𝛾+1. Let 𝑛 be a nonzero fixed natural number and suppose that 𝛾^∗(𝑛) is the smallest integer 𝛾(𝑛) with the property 𝑑𝜉_𝛾,𝜉_𝛾′<13𝑛for𝛾≥𝛾^′≥𝛾(𝑛). (20) Then, there holds 𝛾^∗(𝑛)≥𝛾^∗𝑛^′iff𝑛^′≥𝑛. (21)
For each 𝛾∈ℕ, we also fix a sequence (𝑥_𝑛(𝛾))_{𝑛(𝛾)∈ℕ}∈𝒜_𝜉𝛾. (22) If 𝜉_𝛾=𝜙(𝑥) for some 𝑥∈𝑋, then 𝑥_𝑛(𝛾)=𝑥 for each 𝑛(𝛾)∈ℕ. Let Λ(𝑛)=𝛾^∗(𝑛)+2if𝛾^∗(𝑛)𝛾iseven,_∗(𝑛)+3if𝛾^∗(𝑛)isodd. (23) Then, for each 𝜆≥Λ(𝑛)/2 we have 𝑑𝜉_2𝜆,𝜉_2𝜆−1<1.3𝑛 (24) Hence, since 𝒜_2𝜆−1≠𝒜_2𝜆, there are 𝑛₀(2𝜆)=𝑛₀(𝛾^∗(𝑛),2𝜆)∈ℕ and 𝑛₀(2𝜆−1)∈ℕ such that 𝑑𝑥_𝑛(2𝜆),𝑥_{𝑛(2𝜆−1)}<13𝑛 (25) whenever 𝑛(2𝜆)≥𝑛₀(𝛾^∗(𝑛),2𝜆) and 𝑛(2𝜆−1)≥𝑛₀(2𝜆−1). On the other hand, since (𝑥_𝑛(2𝜆))_{𝑛(2𝜆)∈ℕ} is a right 𝐾-Cauchy sequence, there exists an 𝑛₁(2𝜆)∈ℕ such that 𝑑𝑥_𝑛(2𝜆),𝑥_{𝑛′(2𝜆)}<13𝑛 (26) for each 𝑛(2𝜆)≥𝑛^′(2𝜆)≥𝑛₁(2𝜆).
We put 𝑛(𝛾^∗(𝑛),2𝜆)=max{𝑛₀(𝛾^∗(𝑛),2𝜆),𝑛₁(2𝜆)}. Then, we have 𝑛𝛾^∗(𝑛)≥𝑛𝛾,2𝜆_∗𝑛^′,2𝜆iff𝛾^∗(𝑛)≥𝛾^∗𝑛^′. (27)
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 𝑑𝑥_{𝑛(2𝜆−1)},𝑥_{𝑛(2𝜆′)}<13𝑛 (28) 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 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥𝑛_(2𝜆−1)<1.3𝑛 (29) On the other hand (28) implies that 𝑑𝑥𝑛_(2𝜆−1),𝑥_{𝑛(2𝜆′)}<13𝑛 (30) for each 𝑛(2𝜆^′)≥𝑛₂(2𝜆^′) for some 𝑛₂(2𝜆^′)∈ℕ. Hence, 𝑑𝑥𝑛_(2𝜆−1),𝑥_{𝑛2(2𝜆′)}<1.3𝑛 (31) 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 𝑑𝑥𝑛_(2𝜆−1)𝑛,𝑥_{(𝛾∗(𝑛),2𝜆′)}<1.3𝑛 (32) Hence, 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛,𝑥_{(𝛾∗(𝑛),2𝜆′)}<𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥𝑛_(2𝜆−1)+𝑑𝑥𝑛_(2𝜆−1)𝑛,𝑥_{(𝛾∗(𝑛),2𝜆′)}<1+13𝑛<13𝑛𝑛. (33) In case (ii), we have 𝑛₂(2𝜆^′)≥𝑛(𝛾^∗(𝑛),2𝜆^′)=max{𝑛₀(𝛾^∗(𝑛),2𝜆^′),𝑛₁(2𝜆^′)}; hence (26) implies that 𝑑𝑥_{𝑛2(2𝜆′)}𝑛,𝑥_{(𝛾∗(𝑛),2𝜆′)}<1.3𝑛 (34) Therefore, (29), (30), and (34) imply that 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛,𝑥_{(𝛾∗(𝑛),2𝜆′)}<𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥𝑛_(2𝜆−1)+𝑑𝑥𝑛_(2𝜆−1),𝑥_{𝑛2(2𝜆′)}+𝑑𝑥_{𝑛2(2𝜆′)}𝑛,𝑥_{(𝛾∗(𝑛),2𝜆′)}<1+13𝑛+13𝑛<13𝑛𝑛. (35) Hence, in any case, we have 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛,𝑥_{(𝛾∗(𝑛),2𝜆′)}<1𝑛 (36) for each 𝜆≥𝜆^′≥Λ(𝑛)/2.
We consider the set Θ=𝑛𝛾^∗(𝑛),2𝜆∣𝑛,𝜆∈ℕ. (37) According to (27) we give to Θ the following order: 𝜗^′=𝑛𝛾^∗𝑛^′,2𝜆^′⪷𝜗=𝑛𝛾^∗(𝑛),2𝜆 (38) if and only if 𝜆=𝜆^′,𝛾^∗𝑛^′≤𝛾^∗(𝑛),or𝜆^′<𝜆. (39) 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 𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛_(𝛾^∗_{(𝑛),2𝜆)∈ℕ} (40) is a right 𝐾-Cauchy sequence which satisfies the assumptions of the lemma.
Let 𝜀>0 and 𝑛_𝜀=min{𝑛∣2/𝑛<𝜀}. Suppose that 𝑛𝛾^∗𝑛_𝜀,Λ𝑛_𝜀≤𝑛𝛾^∗𝑛^′,2𝜆^′≤𝑛𝛾^∗(𝑛),2𝜆. (41) We prove that 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛,𝑥_{(𝛾∗(𝑛′),2𝜆′)}<𝜀. (42) If 𝜆^′=𝜆 and 𝛾^∗(𝑛^′)=𝛾^∗(𝑛)), then 𝑛(𝛾^∗(𝑛),2𝜆)=𝑛(𝛾^∗(𝑛^′),2𝜆^′). Therefore, 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛,𝑥_{(𝛾∗(𝑛′),2𝜆′)}=0<𝜀. (43) Suppose that 𝑛(𝛾^∗(𝑛_𝜀),Λ(𝑛_𝜀))≤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^∗(𝑛_𝜀),Λ(𝑛_𝜀))≤𝑛(𝛾^∗(𝑛^′),2𝜆)<𝑛(𝛾^∗(𝑛),2𝜆). Since (𝑥_𝑛(2𝜆))_{𝑛(2𝜆)∈ℕ} is a right 𝐾-Cauchy sequence we have that 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛,𝑥_{(𝛾∗(𝑛′),2𝜆′)}=𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥𝑛_(𝛾^∗_(𝑛^′_),2𝜆)<13𝑛_𝜀<2𝑛_𝜀<𝜀. (44) 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^∗(𝑛_𝜀),Λ(𝑛_𝜀))≤𝑛(𝛾^∗(𝑛^′),2𝜆^′)<𝑛(𝛾^∗(𝑛),2𝜆^′). Since (𝑥_{𝑛(2𝜆′)})_{𝑛(2𝜆′)∈ℕ} is a right 𝐾-Cauchy sequence we have that 𝑑𝑥𝑛_(𝛾^∗_(𝑛),2𝜆^′₎𝑛,𝑥_{(𝛾∗(𝑛′),2𝜆′)}<13𝑛_𝜀. (45) Then, (36) and (45) imply that 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛,𝑥_{(𝛾∗(𝑛′),2𝜆′)}<𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛,𝑥_{(𝛾∗(𝑛),2𝜆′)}𝑥𝑛+𝑑_{(𝛾∗(𝑛),2𝜆′)}𝑛,𝑥_{(𝛾∗(𝑛′),2𝜆′)}<1𝑛_𝜀+13𝑛_𝜀<2𝑛_𝜀<𝜀. (46) In order to prove subcase (II₂), we combine (29) and (30). So, we get 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},x_{𝑛(2𝜆′)}<13𝑛_𝜀+13𝑛_𝜀=23𝑛_𝜀 (47) 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^∗(𝑛_𝜀),Λ(𝑛_𝜀)). In the first subcase, (45) implies that 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥𝑛_(𝛾^∗_(𝑛^′_),2𝜆)<23𝑛_𝜀<2𝑛_𝜀<𝜀. (48) In the second subcase, we have 𝑑𝑥_{𝑛2(2𝜆′)}𝑛,𝑥_{(𝛾∗(𝑛′),2𝜆′)}<13𝑛_𝜀. (49) Therefore, (45) and (49) imply that 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}𝑛,𝑥_{(𝛾∗(𝑛′),2𝜆′)}<𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥_{𝑛2(2𝜆′)}+𝑑𝑥_{𝑛2(2𝜆′)}𝑛,𝑥_{(𝛾∗(𝑛′),2𝜆′)}<23𝑛_𝜀+13𝑛_𝜀<2𝑛_𝜀<𝜀. (50) Hence, in any case the inequality (42) is invalid.
(b)  (𝜙(𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}))𝑛_(𝛾^∗_{(𝑛),2𝜆)∈ℕ}  and  (𝜉_𝛾)_𝛾∈ℕ  Are Right  𝑑-Cofinal. We first show that (𝜉_𝛾)_𝛾∈ℕ is right 𝑑-cofinal to (𝜙(𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}))𝑛_(𝛾^∗_{(𝑛),2𝜆)∈ℕ}. Fix an 𝜀>0 and let 𝑁_𝜀=min{𝑛∣1/3𝑛<𝜀}. Suppose that 𝛾≥Λ(𝑁_𝜀). Then, we define (𝑛(𝛾^∗(𝑛),2𝜆))_𝛾=(𝑛(𝛾^∗(𝑁_𝜀),2𝜆_𝛾), where 2𝜆_𝛾=𝛾+mod(𝛾,2)+2.
We prove that 𝑑𝜙𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝜉_𝛾<𝜀 (51) for each 𝑛(𝛾^∗(𝑛),2𝜆)≥(𝑛(𝛾^∗(𝑁_𝜀),2𝜆_𝛾). That is, if (𝑥_𝑠)_𝑠∈ℕ∈𝒜_𝜉𝛾, then 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥_𝑠<𝜀 (52) for each 𝑠≥𝑠_𝜀 for some 𝑠_𝜀∈ℕ. Indeed, fix a 𝜆∈ℕ such that Λ(𝑁_𝜀)<𝛾<2𝜆_𝛾<2𝜆−1<2𝜆. Then, from (25) which is valid for 𝑛(2𝜆)=𝑛(𝛾^∗(𝑛),2𝜆) we have 𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥_{𝑛(2𝜆−1)}<13𝑁_𝜀 (53) for each 𝑛(2𝜆−1)≥𝑛₀(2𝜆−1). Now, since Λ(𝑁_𝜀)<𝛾<2𝜆−1 we have 𝑥_{𝑛(2𝜆−1)},𝑥_𝑠<13𝑁_𝜀 (54) for some 𝑛(2𝜆−1), 𝑠 onwards. On the other hand, since (𝑥_𝑠)_𝑠∈ℕ is right 𝐾-Cauchy sequence we have 𝑥_𝑠,𝑥_𝑠′<13𝑁_𝜀 (55) for 𝑠≥𝑠^′≥𝑠_𝜀. Finally, by combining the last three inequalities, we obtain 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥_𝑠<13𝑁_𝜀+13𝑁_𝜀+13𝑁_𝜀<1𝑁_𝜀<𝜀 (56) for each 𝑠≥𝑠_𝜀. The last conclusion confirms the truth of (51).
To prove that (𝜉_𝛾)_𝛾∈ℕ is right 𝑑-cofinal to 𝜙(𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)})𝑛_(𝛾^∗_{(𝑛),2𝜆)∈ℕ}, fix an 𝜀>0. We define 𝑛_𝜀=min{𝑛∣2/𝑛<𝜀}. Now, let 𝑛(𝛾^∗(𝑛),2𝜆)≥𝑛(𝛾^∗(𝑛_𝜀),Λ(𝑛_𝜀)) and let 𝛾𝑛_(𝛾^∗_{(𝑛),2𝜆)}=2𝜆+2. We prove that 𝑑𝜉_𝛾,𝜙𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}<13𝑛 (57) for each 𝛾>𝛾𝑛^{(𝛾∗(𝑛),2𝜆)}. Indeed, let (𝑥_𝑠)_𝑠∈ℕ∈𝒜_𝜉𝛾 and (𝑥_𝑡)_𝑡∈ℕ∈𝒜_𝜙(𝑛^{(𝛾∗(𝑛),2𝜆))}. Since 𝑑𝜉_𝛾,𝜉_2𝜆−1<13𝑛_𝜀,𝑑𝜉_2𝜆−1,𝜉_2𝜆<13𝑛_𝜀 (58) using the same logic as above, we conclude that 𝑑𝑥_𝑠,𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}<1𝑛_𝜀 (59) for each 𝑠≥𝑠_𝜀 for some 𝑠_𝜀∈ℕ. On the other hand, since 𝑥_𝑡→𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)} we have 𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥_𝑡<1𝑛_𝜀 (60) whenever 𝑡≥𝑡_𝜀 for some 𝑡_𝜀∈ℕ. Therefore, (59) and (60) imply that 𝑑𝑥_𝑠,𝑥_𝑡<𝑑𝑥_𝑠,𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)}+𝑑𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)},𝑥_𝑡<2𝑛_𝜀<𝜀 (61) for each 𝑠≥𝑠_𝜀 and each 𝑡≥𝑡_𝜀. Finally, (61) implies (57) and therefore the required sequence (𝑥_𝑛)_𝑛∈ℕ of the hypothesis is the sequence (𝑥𝑛_(𝛾^∗_{(𝑛),2𝜆)})𝑛_(𝛾^∗_{(𝑛),2𝜆)∈ℕ}.

Proposition 26. Let (𝜂)=(𝜂_𝛿)_𝛿∈ℕ be a nonconstant left 𝐾-Cauchy sequence of (𝑋,𝑑) without last element. Then there exists a left 𝐾-Cauchy sequence (𝑦_𝜎)_𝜎∈ℕ of (𝑋,𝑑) such that the sequences (𝜂_𝛿)_𝛿∈ℕ and (𝜙(𝑦_𝜎))_𝜎∈ℕ are left 𝑑-cofinal.

Theorem 27. Every quasi-pseudometric space has a 𝜅-completion.

Proof. Let (𝑋,𝑑) be a quasi-pseudometric space and let (𝜉_𝛾)_𝛾∈ℕ be a 𝜅-Cauchy sequence in the space (𝑋,𝑑). Then, by Definition 11, there exists a 𝜅-cut ̂𝜉=(𝒜𝜉̂,ℬ𝜉̂)∈𝑋 such that (𝜉_𝛾)_𝛾∈ℕ̂𝜉∈𝒜. Suppose that 𝒜̂𝜉={(𝜉^𝑖_𝛾(𝑖))_{𝛾(𝑖)∈ℕ}∣𝑖∈𝐼} and ℬ̂𝜉={(𝜂^𝑗_𝛿(𝑗))_{𝛿(𝑗)∈ℕ}∣𝑗∈𝐽}. Then, (𝜉_𝛾)_𝛾∈ℕ=(𝜉^𝑖0_𝛾(𝑖0))_{𝛾(𝑖0)∈ℕ} for some 𝑖₀∈𝐼. We prove that there exists a 𝜅-cut 𝜉 in (𝑋,𝑑) such that (𝜉_𝛾)_𝛾∈ℕ converges to 𝜉 with respect to 𝜏(𝑑).
We define that 𝒜_𝜉={(𝑥_𝑛)_𝑛∈ℕ∣(𝑥_𝑛)_𝑛∈ℕ is a right 𝐾-Cauchy sequence of 𝑋 such that (𝜙(𝑥_𝑛)_𝑛∈ℕ̂𝜉∈𝒜} and ℬ_𝜉={(𝑦_𝜎)_𝜎∈ℕ∣(𝑦_𝜎)_𝜎∈ℕ is a left 𝐾-Cauchy sequence of 𝑋 such that (𝜙(𝑦_𝜎)_𝜎∈ℕ̂𝜉∈ℬ}.
We first verify that 𝜉=(𝒜_𝜉,ℬ_𝜉) constitute a 𝜅-cut in (𝑋,𝑑). For this we need to show that (A)the classes 𝒜_𝜉, ℬ_𝜉 are nonvoid,(B)the pair (𝒜_𝜉,ℬ_𝜉) satisfies the conditions of Definition 8.
We first show (A). We have two cases to consider.
Case A.1. There exists  𝑖̃∈𝐼  such that the sequence  𝑖̃(𝜉_𝛾(̃𝑖₎)_𝛾(𝑖̃_)∈ℕ  is nonconstant. In this case, Proposition 25 implies that there is a right 𝐾-Cauchy sequence (𝑥_𝑛)_𝑛∈ℕ of 𝑋 such that 𝑖̃(𝜉_𝛾(̃𝑖₎)_𝛾(𝑖̃_)∈ℕ and (𝜙(𝑥_𝑛))_𝑛∈ℕ are right 𝑑-cofinal. Therefore, (𝜙(𝑥_𝑛))_𝑛∈ℕ̂𝜉∈𝒜 which implies that (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉.
Case A.2. For each  𝑖∈𝐼, there exists  𝛾(𝑖)∈ℕ  such that  𝜉^𝑖_𝛾(𝑖)=𝜉^𝑖_𝛾(𝑖)+1  whenever  𝛾(𝑖)≥𝛾(𝑖). Fix an 𝑖^∗∈𝐼. Suppose that (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉𝑖^∗𝛾^∗_)(𝑖. Then, from lim_{𝛿(𝑗),𝛾(𝑖}^∗₎𝑑(𝜂^𝑗_𝛿(𝑗),𝜉^𝑖∗_𝛾(𝑖^∗₎)=0,  𝑑(𝜉^𝑖∗_𝛾(𝑖^∗₎,𝜉𝛾^𝑖∗_(𝑖^∗₎)=0, and lim_𝑛𝑑(𝜉𝛾^𝑖∗_(𝑖^∗₎, 𝜙(𝑥_𝑛))=0, we conclude that lim_{𝛿(𝑗),𝑛}𝑑(𝜂^𝑗_𝛿(𝑗),𝜙(𝑥_𝑛))=0. It follows that (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉. Therefore, in any case we have that 𝒜_𝜉≠∅. Similarly we prove that ℬ_𝜉≠∅.
To show (B), we first prove the validity of Condition (i) of Definition 8. Let (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉 and (𝑦_𝜎)_𝜎∈ℕ∈ℬ_𝜉. Then, by construction of 𝒜_𝜉 and ℬ_𝜉, we have lim_𝜎,𝑛𝑑(𝜙(𝑦_𝜎),𝜙(𝑥_𝑛))=0. Hence, Proposition 22 implies that lim_𝜎,𝑛𝑑(𝑦_𝜎,𝑥_𝑛)=0. To prove that 𝜉 satisfies the second condition of Definition 8, let (𝑥_𝑛)_𝑛∈ℕ and (𝑥_𝑚)_𝑚∈ℕ be two right 𝐾-Cauchy sequences of 𝒜_𝜉. Since (𝜙(𝑥_𝑛))_𝑛∈ℕ and (𝜙(𝑥_𝑚))_𝑚∈ℕ belong to 𝒜̂𝜉, it follows by definition that they are right 𝑑-cofinal. Hence, Proposition 22 implies that (𝑥_𝑛)_𝑛∈ℕ and (𝑥_𝑚)_𝑚∈ℕ are right 𝑑-cofinal. Clearly, the maximality of 𝒜̂𝜉 and ℬ̂𝜉 implies the maximality of 𝒜_𝜉 and ℬ_𝜉, respectively. Hence, condition (iii) is satisfied.
It remains to prove that (𝜉_𝛾)_𝛾∈ℕ converges to 𝜉 with respect to 𝜏(𝑑). If (𝜉_𝛾)_𝛾∈ℕ is constant, then, 𝜉_𝛾=𝜉 for some 𝛾∈ℕ onwards. Hence, by the Definition 17, we have that lim_𝛾𝑑(𝜉,𝜉_𝛾)=0. If the sequence (𝜉_𝛾)_𝛾∈ℕ is nonconstant, then there exists a right 𝐾-Cauchy sequence (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉 such that (𝜙(𝑥_𝑛))_𝑡∈ℕ and (𝜉_𝛾)_𝛾∈ℕ are right 𝑑-cofinal. Since (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉, Proposition 23 implies that 𝜙(𝑥_𝑛)→𝜉. So, by Corollary 7 we conclude that 𝜉_𝛾→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 𝒜_𝜉=𝒜_𝜉′ whenever 𝜉,𝜉^′∈𝑋 or 𝜉=𝜙_𝑖(𝑥) and 𝜉^′=𝜙_𝑗(𝑥) for some 𝑖,𝑗∈𝐼_𝑥 whenever 𝜉,𝜉^′∈Φ(𝑥), 𝑥∈𝑋 (see Definition 9). In the following 𝜙_𝑞∶𝑋→𝑋 will denote the isometric quotient map with respect to ≈ previously defined. That is, 𝜉 corresponds to 𝜙_𝑞(𝜉)=Ξ whenever 𝜉∈𝑋. Clearly, 𝜙_𝑞 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^′,𝜉^′₁∈𝑋 and 𝜉≈𝜉^′ and 𝜉₁≈𝜉^′₁. By the triangle inequality we see that 𝑑(𝜉₁,𝜉^′₁)≤𝑑(𝜉₁,𝜉)+𝑑(𝜉,𝜉^′)+𝑑(𝜉^′,𝜉^′₁) and hence 𝑑(𝜉₁,𝜉^′₁)≤0+𝑑(𝜉,𝜉^′)+0. Similarly, 𝑑(𝜉,𝜉^′)≤𝑑(𝜉₁,𝜉^′₁) which implies that 𝑑(𝜉,𝜉^′)=𝑑(𝜉₁,𝜉^′₁). 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 (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉 and (𝑥^′_𝑚)_𝑚∈ℕ∈𝒜_𝜉′. Then, by Proposition 18 we conclude that (𝑥_𝑛)_𝑛∈ℕ and (𝑥^′_𝑚)_𝑚∈ℕ are right 𝑑-cofinal. Therefore, 𝒜_𝜉=𝒜_𝜉′. 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=𝜙_𝑞∘𝜙∶(𝑋,𝑑)→(𝑋,𝑑) is an isometric embedding.

Proof. We have that 𝜑=𝜙_𝑞∘𝜙 is an isometry, since both 𝜙_𝑞 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 (Ξ_𝛾)_𝛾∈ℕ be a 𝜅-Cauchy sequence in (𝑋,𝑑). Therefore, since 𝜙_𝑞 is a surjective isometry, 𝜙_𝑞⁻¹((Ξ_𝛾)_𝛾∈ℕ) is a 𝜅-Cauchy sequence in (𝑋,𝑑). Thus for some 𝜉∈𝑋, 𝜙_𝑞⁻¹((Ξ_𝛾)_𝛾∈ℕ) converges to 𝜉. It follows that (Ξ_𝛾)_𝛾∈ℕ converges to 𝜙_𝑞(𝜉)=Ξ since 𝜙_𝑞 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)=(𝜙_𝑞∘𝜙)(𝑋) is dense in (𝑋,𝑑). Indeed, suppose that Ξ∈𝑋. Then, since 𝜙_𝑞 is a surjective isometry, there exists 𝜉∈𝑋 such that 𝜙_𝑞(𝜉)=Ξ. Let (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉. Then, by Definition 17 we have that 𝜙(𝑥_𝑛)→𝜉. Therefore, 𝜙_𝑞(𝜙(𝑥_𝑛)) converges to 𝜙_𝑞(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 the κ-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, the 𝜅-completion of the 𝜅-completion of (𝑋,𝑑) is isomorphic to the 𝜅-completion of (𝑋,𝑑).

The 𝜅-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 the 𝜅-completion satisfies the requirements (i) and (ii) previously stated.

Proposition 33. Every convergent right 𝐾-Cauchy sequence is a 𝜅-Cauchy sequence.

Proof. In fact, if a right 𝐾-Cauchy sequence (𝑥_𝑛)_𝑛∈ℕ converges to a point 𝑥 of 𝑋, then it belongs to the first class of a 𝜅-cut 𝜙_𝑖(𝑥)∈Φ(𝑥). The validity of this proposition follows from Definition 11.

Clearly, in the case of metric spaces, the notions of converging right 𝐾-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 (𝑥_𝑛)_𝑛∈ℕ converges to a point, say 𝑥, then (𝑥_𝑛)_𝑛∈ℕ has 𝑥 as its cosequence, so it is a 𝐷-Cauchy sequence. (A sequence (𝑥_𝑛)_𝑛∈ℕ in the quasimetric space (𝑋,𝑑) is called 𝐷-Cauchy sequence provided that for any 𝜅∈ℕ there exists a 𝑦_𝜅∈𝑋 and 𝑁_𝜅∈ℕ such that 𝑑(𝑦_𝑚,𝑥_𝑛)<1/𝜅 when 𝑚,𝑛>𝑁_𝜅.) 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={𝑥_𝑛∣𝑛=0,1,2,…} and 𝑑𝑥_𝑛,𝑥_𝑚=11for𝑚>0,𝑚≠𝑛,𝑛for𝑚=0,𝑛≠0,0for𝑚=𝑛. (62) The sequence (𝑥_𝑛)_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 (𝑥_𝑛)_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, (𝑥_𝑛)_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 of 𝐷-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 category 𝒬𝒫ℳ𝒮 whose objects are the 𝑇₀ quasi-pseudo-metric spaces and morphisms are the 𝜅-cut preserving functions defined below (Definition 39).

Definition 36. A function 𝑓 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 (𝑥_𝑛)_𝑛∈ℕ and (𝑦_𝜎)_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 (𝑥_𝑛)_𝑛∈ℕ∈𝒜 and (𝑦_𝜎)_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 ((𝑥_𝑛)_𝑛∈ℕ, (𝑦_𝜎)_𝜎∈ℕ) be a 𝜅-Cauchy pair of sequences in (𝑋,𝑑). We say that a 𝜅-Cauchy pair ((𝑥_𝑚)_𝑚∈ℕ, (𝑦_𝑠)_𝑠∈ℕ) of sequences in (𝑋,𝑑) is (𝜅,𝑓)-compatible to ((𝑥_𝑛)_𝑛∈ℕ, (𝑦_𝜎)_𝜎∈ℕ) if it satisfies the following conditions:(i)((𝑥_𝑛)_𝑛∈ℕ,(𝑦_𝜎)_𝜎∈ℕ) and ((𝑥_𝑚)_𝑚∈ℕ,(𝑦_𝑠)_𝑠∈ℕ) are 𝜅-compatible;(ii)((𝑓(𝑥_𝑚))_𝑚∈ℕ, (𝑓(𝑦_𝑠))_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 each 𝜅-Cauchy pair ((𝑥_𝑛)_𝑛∈ℕ,(𝑦_𝜎)_𝜎∈ℕ) of sequences in 𝑋, there exists a 𝜅-Cauchy pair ((𝑥_𝑚)_𝑚∈ℕ,(𝑦_𝑠)_𝑠∈ℕ) of sequences in 𝑋 such that ((𝑥_𝑚)_𝑚∈ℕ,(𝑦_𝑠)_𝑠∈ℕ) is (𝜅,𝑓)-compatible to ((𝑥_𝑛)_𝑛∈ℕ,(𝑦_𝜎)_𝜎∈ℕ).(ii)If ((𝑥_𝑛)_𝑛∈ℕ,(𝑦_𝜎)_𝜎∈ℕ),((𝑥^′_𝑛)_𝑛∈ℕ,(𝑦^′_𝜎)_𝜎∈ℕ), ((𝑥_𝑚)_𝑚∈ℕ,(𝑦_𝑠)_𝑠∈ℕ) and ((𝑥^′_𝑚)_𝑚∈ℕ,(𝑦^′_𝑠)_𝑠∈ℕ) are 𝜅-Cauchy pairs of sequences such that ((𝑥_𝑚)_𝑚∈ℕ,(𝑦_𝑠)_𝑠∈ℕ) is (𝜅,𝑓)-compatible to ((𝑥_𝑛)_𝑛∈ℕ,(𝑦_𝜎)_𝜎∈ℕ) and ((𝑥^′_𝑚)_𝑚∈ℕ,(𝑦^′_𝑠)_𝑠∈ℕ) is (𝜅,𝑓)-compatible to ((𝑥^′_𝑛)_𝑛∈ℕ,(𝑦^′_𝜎)_𝜎∈ℕ), then, the 𝜅-compatibility (resp. non-𝜅-compatibility) of ((𝑥_𝑛)_𝑛∈ℕ,(𝑦_𝜎)_𝜎∈ℕ) and ((𝑥^′_𝑛)_𝑛∈ℕ,(𝑦^′_𝜎)_𝜎∈ℕ) implies the 𝜅-compatibility (non-𝜅-compatibility) of ((𝑓(𝑥_𝑚))_𝑚∈ℕ,(𝑓(𝑦_𝑠))_𝑠∈ℕ) and ((𝑓(𝑥^′_𝑚))_𝑚∈ℕ,(𝑓(𝑦^′_𝑠))_𝑠∈ℕ), respectively.

By the previously mentioned definition, it is clear that a cut preserving function 𝑓 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 a 𝑇₀ 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 any 𝜅-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 mapping 𝜑 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)=lim_𝑛𝑓(𝑥_𝑛) whenever 𝜉 is a 𝜅-cut in 𝑋 such that 𝜙_𝑞(𝜉)=Ξ and (𝑥_𝑛)_𝑛∈ℕ∈𝒜_𝜉. Definitions 28 and 39 imply that 𝑓^∗ takes distinct elements of 𝑋 to distinct elements of 𝑌. Let Ξ=𝜑(𝑥) where 𝜑=𝜙_𝑞∘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 ((Ξ_𝛾)_𝛾∈ℕ,(𝐻_𝛽)_𝛽∈ℕ) be a pair Cauchy of sequences in 𝑋. Suppose that for each 𝛾,𝛽∈ℕ we have Ξ_𝛾=𝜙_𝑞(𝜉_𝛾) and 𝐻_𝛽=𝜙_𝑞(𝜂_𝛽). Then, let ((𝜉_𝛾)_𝛾∈ℕ,(𝜂_𝛽)_𝛽∈ℕ) be a pair Cauchy of sequences in 𝑋. Then, by Propositions 25 and 26 and Theorem 27 there is a pair Cauchy ((𝑥_𝑛)_𝑛∈ℕ, (𝑦_𝜎)_𝜎∈ℕ) of sequences in 𝑋 such that ((𝜉_𝛾)_𝛾∈ℕ,(𝜂_𝛽)_𝛽∈ℕ) and ((𝜙(𝑥_𝑛))_𝑛∈ℕ, (𝜙(𝑦_𝜎))_𝜎∈ℕ) are 𝜅-compatible, (𝜉)_𝛾∈ℕ and ((𝜙(𝑥_𝑛))_𝑛∈ℕ are right 𝑑-cofinal, and (𝜂_𝛽)_𝛽∈ℕ and ((𝜙(𝑦_𝜎))_𝜎∈ℕ are left 𝑑-cofinal. Since 𝑓 is cut preserving, there exists a Cauchy pair ((𝑥_𝑚)_𝑚∈ℕ, (𝑦_𝑠)_𝑠∈ℕ) of sequences satisfying the conditions (i) and (ii) of Definition 39. Since 𝜑=𝜙_𝑞∘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 standard 𝜅-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 the 𝜅-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 𝜑_𝑋∶(𝑋,𝑑)→(𝑋,𝑑) and 𝜑_𝑌∶(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 a 𝜅-cut preserving function 𝑓^∗∶(𝑋,𝑑)→(𝑌,𝜌) follows from Proposition 40 when one applies it to the function 𝜑_𝑌∘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 the 𝜅-completion). In the following (𝑋,𝑑) denotes the 𝜅-completion of the 𝜅-completion (𝑋,𝑑).

Definition 43. The 𝜅-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. The 𝜅-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={(Ξ^𝑖_𝛾(𝑖))_{𝛾(𝑖)∈ℕ}∣𝑖∈𝐼} and ℬ̃𝜉={(𝐻^𝑗_𝛿(𝑗))_{𝛿(𝑗)∈ℕ}∣𝑗∈𝐽}. Then, the families 𝒜̂𝜉={(𝜉^𝑖_𝛾(𝑖))_{𝛾(𝑖)∈ℕ}∣𝑖∈𝐼} and ℬ̂𝜉={(𝜂^𝑗_𝛿(𝑗))_{𝛿(𝑗)∈ℕ}∣𝑗∈𝐽}, where 𝜙_𝑞(𝜉^𝑖_𝛾(𝑖))=Ξ^𝑖_𝛾(𝑖) and 𝜙_𝑞(𝜂^𝑗_𝛿(𝑗))=𝐻^𝑗_𝛿(𝑗), define a 𝜅-cut ̂𝜉=(𝒜𝜉̂,ℬ𝜉̂) in 𝑋. Then, according to Theorem 27, to each 𝜅-cut 𝜉̂ there corresponds a 𝜅-cut 𝜉∈(𝑋,𝑑). Let Ξ=𝜙_𝑞(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 𝑑Ξ^′,Ξ^′′=𝑑𝜑Ξ^′,𝜑Ξ^′′=𝑑Ξ^′,Ξ^′′=𝑑𝜉^′,𝜉^′′, (63) where Ξ^′=𝜙_𝑞(𝜉^′) and Ξ^′′=𝜙_𝑞(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 ΞΞΞ=(𝒜,ℬ) in (𝑋,𝑑) such that (Ξ_𝛾)_𝛾∈ℕΞ∈𝒜. Suppose that 𝒜Ξ={(Ξ^𝑖_𝛾(𝑖))_{𝛾(𝑖)∈ℕ}∣𝑖∈𝐼}, ℬΞ={(𝐻^𝑗_𝛿(𝑗))_{𝛿(𝑗)∈ℕ}∣𝑗∈𝐽} and (Ξ_𝛾)_𝛾∈ℕ=(Ξ^𝑖0_𝛾(𝑖0))_{𝛾(𝑖0)∈ℕ} for some 𝑖₀∈𝐼. For each 𝑖∈𝐼 and each 𝑗∈𝐽 we have Ξ^𝑖_𝛾(𝑖)=𝜑(Ξ^𝑖_𝛾(𝑖)) and 𝐻^𝑖_𝛿(𝑗)=𝜑(𝐻^𝑗_𝛿(𝑗)). On the other hand, Ξ^𝑖_𝛾(𝑖)=𝜙_𝑞(𝜉^𝑖_𝛾(𝑖)) and 𝐻^𝑗_𝛿(𝑗)=𝜙_𝑞(𝜂^𝑗_𝛿(𝑗)) for some 𝜉^𝑖_𝛾(𝑖),𝜂^𝑗_𝛿(𝑗)∈(𝑋,𝑑). Next, we define ̂𝜉=(𝒜𝜉̂,ℬ𝜉̂), where 𝒜̂𝜉={(𝜉^𝑖_𝛾(𝑖))_{𝛾(𝑖)∈ℕ}∣𝑖∈𝐼} and ℬ̂𝜉={(𝜂^𝑖_𝛿(𝑗))_{𝛿(𝑗)∈ℕ}∣𝑗∈𝐽}. By Theorem 27, there exists a 𝜅-cut 𝜉∈(𝑋,𝑑) such that, for each 𝛾(𝑖)∈ℕ, (𝜉^𝑖_𝛾(𝑖))_{𝛾(𝑖)∈ℕ}→𝜉 with respect to 𝜏(𝑑). By definition of 𝑑, it follows that (Ξ^𝑖_𝛾(𝑖))_{𝛾(𝑖)∈ℕ}→Ξ with respect to 𝜏(𝑑), where Ξ=𝜙_𝑞(𝜉). Since 𝜑 is an isometry, Proposition 25 implies that (Ξ^𝑖_𝛾(𝑖))_{𝛾(𝑖)∈ℕ} and (𝜑(Ξ^𝑖_𝛾(𝑖)))_{𝛾(𝑖)∈ℕ} are right 𝑑-cofinal. Therefore, Corollary 7 implies that (Ξ^𝑖_𝛾(𝑖))_{𝛾(𝑖)∈ℕ} converges to 𝜑(Ξ) with respect to 𝜏(𝑑). It follows that (Ξ_𝛾)_𝛾∈ℕ=(Ξ^𝑖0_𝛾(𝑖0))_{𝛾(𝑖0)∈ℕ} converges to 𝜑(Ξ) with respect to 𝜏(𝑑).