Completeness in Quasi-Pseudometric Spaces—A Survey

: The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by ﬁlters in the setting of quasi-metric spaces. We propose a new deﬁnition of right K -Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. 17 (1967), 226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., 49 (1984), 36. A discussion on nets deﬁned over ordered or pre-ordered directed sets is also included.


Introduction
It is well known that completeness is an essential tool in the study of metric spaces, particularly for fixed points results in such spaces. The study of completeness in quasi-metric spaces is considerably more involved, due to the lack of symmetry of the distance-there are several notions of completeness all agreing with the usual one in the metric case (see [1] or [2]). Again these notions are essential in proving fixed point results in quasi-metric spaces as it is shown by some papers on this topic as, for instance, [3][4][5][6] (see also the book [7]). A survey on the relations between completeness and the existence of fixed points in various circumstances is given in [8].
It is known that in the metric case the notions of completeness by sequences, by nets and by filters all agree and, further, the completeness of a metric space is equivalent to the completeness of the associated uniform space (in all senses). In the present paper we show that, in the quasi-metric case, these notions agree in some cases, but there are situations when they are different, mainly for the notions of right-completeness. Such a situation was emphasized by Stoltenberg [9], who proposed a notion of Cauchy net for which completeness agrees with the sequential completeness. Later Gregori and Ferrer [10] found a gap in the proof given by Stoltenberg and proposed a new version of right K-Cauchy net for which sequential completeness and completeness by nets agree. In the present paper we include a discussion on this Gregori-Ferrer notion of Cauchy net and complete these results by proposing a notion of right-K-Cauchy net for which the equivalence with sequential completeness holds.

Metric and Uniform Spaces
For a mapping d : X × X → R on a set X consider the following conditions: respectively. A filter on a set X is a nonempty family F of nonempty subsets of X satisfying the conditions It is obvious that (F2) implies for all n ∈ N and F 1 , . . . , F n ∈ F . A base of a filter F is a subset B of F such that every F ∈ F contains a B ∈ B. A nonempty family B of nonempty subsets of X such that generates a filter F (B) given by A family B satisfying (BF1) is called a filter base.
A relation ≤ on a set X is called a preorder if it is reflexive and transitive, that is, for all x, y, z ∈ X. If the relation ≤ is further antireflexive, i.e.
for all x, y ∈ X, then ≤ is called an order. A set X equipped with a preorder (order) ≤ is called a preordered (ordered) set, denoted as (X, ≤). A preordered set (I, ≤) is called directed if for every i 1 , i 2 ∈ I there exists j ∈ I with i 1 ≤ j and i 2 ≤ j. A net in a set X is a mapping φ : I → X, where (I, ≤) is a directed set. The alternative notation (x i : i ∈ I), where x i = φ(i), i ∈ I, is also used.
A uniformity on a set X is a filter U on X × X such that ∃y ∈ X, (x, y) ∈ M and (y, z) ∈ N}, and The sets in U are called entourages. A base for a uniformity U is a base of the filter U . The composition V • V is denoted sometimes simply by V 2 . Since every entourage contains the diagonal ∆(X), the inclusion For U ∈ U , x ∈ X and Z ⊆ X put A uniformity U generates a topology τ(U ) on X for which the family of sets {U(x) : U ∈ U } is a base of neighborhoods of any point x ∈ X.
Let (X, d) be a pseudometric space. Then the pseudometric d generates a topology τ d for which {B d (x, r) : r > 0}, is a base of neighborhoods for every x ∈ X.
The pseudometric d generates also a uniform structure U d on X having as basis of entourages the sets it follows that the topology τ(U d ) agrees with the topology τ d generated by the pseudometric d.
A sequence (x n ) in (X, d) is called Cauchy (or fundamental) if for every ε > 0 there exists n ε ∈ N such that d(x n , x m ) < ε for all m, n ∈ N with m, n n ε , a condition written also as lim m,n→∞ d(x m , x n ) = 0 .
A sequence (x n ) in a uniform space (X, U ) is called U -Cauchy (or simply Cauchy) if for every U ∈ U there exists n ε ∈ N such that (x m , x n ) ∈ U for all m, n ∈ N with m, n n ε .
It is obvious that, in the case of a pseudometric space (X, d), a sequence is Cauchy with respect to the pseudometric d if and only if it is Cauchy with respect to the uniformity U d .
The Cauchyness of nets in pseudometric and in uniform spaces is defined by analogy with that of sequences.
A filter F in a uniform space (X, U ) is called U -Cauchy (or simply Cauchy) if for every U ∈ U there exists F ∈ F such that F × F ⊆ U .

Definition 1.
A pseudometric space (X, d) is called complete if every Cauchy sequence in X converges. A uniform space (X, U ) is called sequentially complete if every U -Cauchy sequence in X converges and complete if every U -Cauchy net in X converges (or, equivalently, if every U -Cauchy filter in X converges).

Remark 1.
We can define the completeness of a subset Y of a pseudometric space (X, d) by the condition that every Cauchy sequence in Y converges to some element of Y. A closed subset of a pseudometric space is complete and a complete subset of a metric space is closed. A complete subset of a pseudometric space need not be closed.
The following result holds in the metric case.
Theorem 1. For a pseudometric space (X, d) the following conditions are equivalent.
1. The metric space X is complete.
2. Every Cauchy net in X is convergent.
3. Every Cauchy filter in (X, U d ) is convergent.
An important result in metric spaces is Cantor characterization of completeness.
Theorem 2 (Cantor theorem). A pseudometric space (X, d) is complete if and only if every descending sequence of nonempty closed subsets of X with diameters tending to zero has nonempty intersection. This means that for any family F n , n ∈ N, of nonempty closed subsets of X If d is a metric then this intersection contains exactly one point.

Quasi-Pseudometric and Quasi-Uniform Spaces
In this section we present the basic results on quasi-metric and quasi-uniform spaces needed in the sequel, using as basic source the book [2].

Quasi-Pseudometric Spaces
Dropping the symmetry condition (M2) in the definition of a metric one obtains the notion of quasi-pseudometric, that is, a quasi-pseudometric on an arbitrary set X is a mapping d : X × X → R satisfying the conditions (M1) and (M3). If d satisfies further (M4 ) then it called a quasi-metric. The pair (X, d) is called a quasi-pseudometric space, respectively a quasi-metric space (In [2] the term "quasi-semimetric" is used instead of "quasi-pseudometric"). Quasi-pseudometric spaces were introduced by Wilson [11].
The conjugate of the quasi-pseudometric d is the quasi-pseudometricd(x, y) = d(y, x), x, y ∈ X. The mapping d s (x, y) = max{d(x, y),d(x, y)}, x, y ∈ X, is a pseudometric on X which is a metric if and only if d is a quasi-metric.
If (X, d) is a quasi-pseudometric space, then for x ∈ X and r > 0 we define the balls in X by (1) The topology τ d (or τ(d)) of a quasi-pseudometric space (X, d) can be defined through the family V d (x) of neighborhoods of an arbitrary point x ∈ X: The topological notions corresponding to d will be prefixed by d-(e.g., d-closure, d-open, etc.). The convergence of a sequence (x n ) to x with respect to τ d , called d-convergence and denoted by x n d − → x, can be characterized in the following way Also As a space equipped with two topologies, τ d and τd , a quasi-pseudometric space can be viewed as a bitopological space in the sense of Kelly [12].
The following example of quasi-metric space is an important source of counterexamples in topology (see [13]).  Asymmetric normed spaces Let X be a real vector space. A mapping p : X → R is called an asymmetric seminorm on X if for all x, y ∈ X and α 0. If, further, for all x ∈ X, then p is called an asymmetric norm.
To an asymmetric seminorm p one associates a quasi-pseudometric d p given by which is a quasi-metric if p is an asymmetric norm. All the topological and metric notions in an asymmetric normed space are understood as those corresponding to this quasi-pseudometric d p (see [2]). The following asymmetric norm on R is essential in the study of asymmetric normed spaces (see [2]).
In this space the addition is continuous from is equipped with the topology generated by | · | and R with τ u .
The following topological properties are true for quasi-pseudometric spaces.
Recall that a topological space (X, τ) is called: • T 0 if, for every pair of distinct points in X, at least one of them has a neighborhood not containing the other; • T 1 if, for every pair of distinct points in X, each of them has a neighborhood not containing the other; • T 2 (or Hausdorff) if every two distinct points in X admit disjoint neighborhoods; • regular if, for every point x ∈ X and closed set A not containing x, there exist the disjoint open sets U, V such that x ∈ U and A ⊆ V.

Remark 2.
It is known that the topology τ d of a pseudometric space (X, d) is Hausdorff (or T 2 ) if and only if d is a metric if and only if any sequence in X has at most one limit. The characterization of Hausdorff property of quasi-pseudometric spaces can also be given in terms of uniqueness of the limits of sequences, as in the metric case: the topology of a quasi-pseudometric space (X, d) is Hausdorff if and only if every sequence in X has at most one d-limit if and only if every sequence in X has at most oned-limit (see [11]).
In the case of an asymmetric seminormed space there exists a characterization in terms of the asymmetric seminorm (see [2], Proposition 1.1.40).

Quasi-Uniform Spaces
Again, the notion of quasi-uniform space is obtained by dropping the symmetry condition (U3) from the definition of a uniform space, that is, a quasi-uniformity on a set X is a filter U in X × X satisfying the conditions (U1) and (U2). The sets in U are called entourages and the pair (X, U ) is called a quasi-uniform space, as in the case of uniform spaces.
As uniformities, a quasi-uniformity U generates a topology τ(U ) on X in a similar way: the sets form a base of neighborhoods of any point x ∈ X. The topology τ(U ) is T 0 if and only if U is a partial order on X, and T 1 if and only if U = ∆(X).
The family of sets is another quasi-uniformity on X called the quasi-uniformity conjugate to U . Also U ∪ U −1 is a subbase of a uniformity U s on X, called the associated uniformity to the quasi-uniformity U . It is the coarsest uniformity on X finer than both U and U −1 , U s = U ∨ U −1 . A basis for U s is formed by the sets is a basis for a quasi-uniformity U d on X. The family , it follows that the topologies generated by the quasi-pseudometric d and by the quasi-uniformity U d agree, i.e., τ d = τ(U d ).
In this case Remark 3. Quasi-uniform spaces were studied by Nachbin starting with 1948 (see [14]) as a generalization of uniform spaces introduced by Weil [15]. For further developments see [16,17].

Cauchy Sequences and Sequential Completeness in Quasi-Pseudometric and Quasi-Uniform Spaces
The lost of symmetry causes a lot of trouble in the study of quasi-metric spaces, particularly concerning completeness and compactness. Starting from the definition of a Cauchy sequence in a metric space, Reilly et al. [1] defined 7 kinds of Cauchy sequences, yielding 14 different notions of completeness in quasi-metric spaces, all agreeing with the usual one in the metric case. One of the major drawbacks of most of these notions is that a convergent sequence need not be Cauchy. For a detailed study we refer to the quoted paper by Reilly et al., or to the book [2]. In the present paper we concentrate on the relations between the corresponding notions of completeness by sequences, nets or filters, as well as to the completeness of the associated quasi-uniform space.
We give now the definitions following [1].
for all n n 0 ; • d s -Cauchy if it is a Cauchy sequence is the pseudometric space (X, d s ), that is for every ε > 0 there exists n 0 ∈ N such that d s (x n , x k ) < ε for all n, k n 0 , a condition equivalent to d(x n , x k ) < ε for all n, k n 0 , as well as tod (x n , x k ) < ε for all n, k n 0 ; for all n, k ∈ N with n 0 k n; for all n n 0 .
Sometimes, to emphasize the quasi-pseudometric d, we shall say that a sequence is left d-K-Cauchy, etc.
It seems that K in the definition of a left K-Cauchy sequence comes from Kelly [12] who considered first this notion.
Some remarks are in order.
1. These notions are related in the following way: We also mention the following simple properties of Cauchy sequences.
To each of these notions of Cauchy sequence corresponds two notions of sequential completeness, by asking that the corresponding Cauchy sequence be d-convergent or d s -convergent. Due to the equivalence d-left Cauchy ⇐⇒d-right Cauchy one obtains nothing new by asking that a d-left Cauchy sequence isd-convergent. For instance, thed-convergence of any left d-K-Cauchy sequence is equivalent to the right K-completeness of the space (X,d). • bicomplete if the associated pseudometric space (X, d s ) is complete, i.e., every d s -Cauchy sequence is d s -convergent. A bicomplete asymmetric normed space (X, p) is called a biBanach space.
As we noticed (see Remark 4 (4)), each d-convergent sequence is left d-Cauchy, but for each of the other notions there are examples of d-convergent sequences that are not Cauchy, which is a major inconvenience. Another one is that a complete (in some sense) subspace of a quasi-metric space need not be closed.
The implications between these completeness notions are obtained by reversing the implications between the corresponding notions of Cauchy sequence from Remark 4 (1). No one of the above implication is reversible (see [1]), excepting that between weakly left and left K-sequential completeness, as it was surprisingly shown by Romaguera [21]. A series ∑ n x n in an asymmetric seminormed space (X, p) is called convergent if there exists x ∈ X such that x = lim n→∞ ∑ n k=1 x k (i.e., lim n→∞ p (∑ n k=1 x k − x) = 0). The series ∑ n x n is called absolutely convergent if ∑ ∞ n=1 p(x n ) < ∞. It is well-known that a normed space is complete if and only if every absolutely convergent series is convergent. A similar result holds in the asymmetric case too.
3. An asymmetric seminormed space (X, p) is sequentially left K-complete if and only if every absolutely convergent series is convergent.

Cantor type results
Concerning Cantor-type characterizations of completeness in terms of descending sequences of closed sets (the analog of Theorem 2) we mention the following result. The diameter of a subset A of a quasi-pseudometric space (X, d) is defined by diam(A) = sup{d(x, y) : x, y ∈ A} .
It is clear that, as defined, the diameter is, in fact, the diameter with respect to the associated pseudometric d s . Recall that a quasi-pseudometric space is called sequentially d-complete if every d s -Cauchy sequence is d-convergent (see Definition 3).

Theorem 3 ([1], Theorem 10).
A quasi-pseudometric space (X, d) is sequentially d-complete if and only if each decreasing sequence F 1 ⊇ F 2 . . . of nonempty closed sets with diam(F n ) → 0 as n → ∞ has nonempty intersection, which is a singleton if d is a quasi-metric.
The following characterization of right K-completeness was obtained in [22], using a different terminology.
If the topology d is Hausdorff, then ∞ n=1 Bd[x n , r n ] contains exactly one element.

Completeness by Nets and Filters
In this section we shall examine the relations between completeness by sequences, nets and filters in quasi-metric spaces. For some notions of completeness they agree, but, as it was shown by Stoltenberg [9], they can be different for others.

Some Positive Results
These hold mainly for the notions of left-completeness, and may fail for those of right completeness as we shall see in the next subsection.
The Cauchy properties of a net (x i : i ∈ I) in a quasi-pseudometric space (X, d) are defined by analogy with that of sequences, by replacing in Definition 2 the natural numbers with the elements of the directed set I.
The situation is good for left Smyth completeness (see Definition 3).

Every left d-K-Cauchy sequence is d s -convergent. 2. Every left d-K-Cauchy net is d s -convergent.
A quasi-uniform space (X, U ) is called bicomplete if (X, U s ) is a complete uniform space. This notion is useful and easy to handle, because one can appeal to well known results from the theory of uniform spaces, but it is not appropriate for the study of the specific properties of quasi-uniform spaces, so one introduces adequate definitions, by analogy with quasi-pseudometric spaces.

Definition 4. Let (X, U ) be a quasi-uniform space.
A filter F on (X, U ) is called: A net (x i : i ∈ I) in (X, U ) is called: • left U -Cauchy (right U -Cauchy) if for every U ∈ U there exists x ∈ X and i 0 ∈ I such that (x, The notions of left and right U -K-Cauchy filter were defined by Romaguera in [21].
so that a filter is right U -K-Cauchy if and only if it is left U −1 -K-Cauchy. A similar remark applies to U -nets.

Definition 5.
A quasi-uniform space (X, U ) is called: • left U -complete by filters (left K-complete by filters) if every left U -Cauchy (respectively, left U -K-Cauchy) filter in X is τ(U )-convergent; • left U -complete by nets (left U -K-complete by nets) if every left U -Cauchy (respectively, left U -K-Cauchy) net in X is τ(U )-convergent; • Smyth left U -K-complete by nets if every left K-Cauchy net in X is U s -convergent.
The notions of right completeness are defined similarly, by asking the τ(U )-convergence of the corresponding right Cauchy filter (or net) with respect to the topology τ(U ) (or with respect to τ(U s ) in the case of Smyth completeness).
As we have mentioned in Introduction, in pseudometric spaces the sequential completeness is equivalent to the completeness defined in terms of filters, or of nets. Romaguera [21] proved a similar result for the left K-completeness in quasi-pseudometric spaces.

Remark 6.
In the case of a quasi-pseudometric space the considered notions take the following form.
A filter F in a quasi-pseudometric space (X, d) is called left K-Cauchy if it left U d -K-Cauchy. This is equivalent to the fact that for every ε > 0 there exists F ε ∈ F such that Proposition 7 ([21]). For a quasi-pseudometric space (X, d) the following are equivalent.
1. The space (X, d) is sequentially left K-complete.

Every left K-Cauchy net in X is d-convergent.
In the case of left U d -completeness this equivalence does not hold in general.

Proposition 8 (Künzi [23]). A Hausdorff quasi-metric space (X, d) is sequentially left d-complete if and only
if the associated quasi-uniform space (X, U d ) is left U d -complete by filters.

Right K-Completeness in Quasi-Pseudometric Spaces
It is strange that for the right completeness the things look worse than for the left completeness. As remarked Stoltenberg [9] (Example 2.4), a result similar to Proposition 7 does not hold for right K-completeness: there exists a sequentially right K-complete T 1 quasi-metric space which is not right K-complete by nets. Actually, Stoltenberg [9] proved that the equivalence holds for a more general definition of a right K-Cauchy net, but Gregori and Ferrer [10] found a gap in Stoltenberg's proof and proposed a new version of Cauchy net. In what follows we will present these results and, in our turn, we shall propose other notion of Cauchy net for which the equivalence holds.
An analog of Proposition 7 for right K-completeness can be obtained only under some supplementary hypotheses on the quasi-pseudometric space X.
A quasi-pseudometric space (X, d) is called R 1 if for all x, y ∈ X, d-cl{x} = d-cl{y} implies the existence of two disjoint d-open sets U, V such that x ∈ U and y ∈ V.
1. If X is right K-complete by filters, then every right K-Cauchy net in X is convergent. In particular, every right K-complete by filters quasi-pseudometric space is sequentially right K-complete. 2. If the quasi-pseudometric space (X, d) is R 1 then X is right K-complete by filters if and only if it is sequentially right K-complete.

Stoltenberg's example
As we have mentioned, Stoltenberg [9] (Example 2.4), gave an example of a sequentially right K-complete T 1 quasi-metric space which is not right K-complete by nets, which we shall present now.
Denote by A the family of all countable subsets of the interval [0, 1 3 ]. For A ∈ A let otherwise .
Proposition 10. (S, d) is a sequentially right K-complete T 1 quasi-metric space which is not right K-complete by nets.
Proof. The proof that d is a T 1 quasi-metric on S is straightforward.
Let (X n ) n∈N be a right K-Cauchy sequence in S. Then there exists n 0 ∈ N such that d(X m , X n ) < 1 for all m, n ∈ N with n 0 n m .
it follows k i ∈ N for all i ∈ N 0 . For 0 < ε < 1 there exists i 0 ∈ N such that d(X n 0 +i+1 , X n 0 +i ) < ε for all i i 0 , which means that This shows that lim i→∞ k i = ∞.
which shows that the sequence (X n ) is d-convergent to X.
II. The quasi-metric space (S, d) is not right K-complete by nets.
We have for X A i , X B j ∈ S 0 , (S 0 , ) is directed and the mapping φ : S 0 → S defined by φ(X) = X, X ∈ S 0 , is a net in S.
Let us show first that the net φ is right K-Cauchy. For ε > 0 choose k ∈ N such that 2 −k < ε. For some C ∈ A, X C k belongs to S 0 and showing that the net φ is right K-Cauchy. Let X = X C k be an arbitrary element in S. We show that for every X A i ∈ S 0 there exists X B j ∈ S 0 with X A i X B j such that d(X, X B j ) = 1, which will imply that the net φ is not d-convergent to X. Since C is a countable set, there exists x 0 ∈ [0, 1 3 i and X C k X B i , so that, by the definition of the metric d, d(X C k , X B i ) = 1.

Stoltenberg-Cauchy nets
Stoltenberg [9] also considered a more general definition of a right K-Cauchy net as a net (x i : i ∈ I) satisfying the condition: for every ε > 0 there exists i ε ∈ I such that Let us call such a net Stoltenberg-Cauchy and Stoltenberg completeness the completeness with respect to Stoltenberg-Cauchy nets.
It follows that, for this definition, where i j means that i, j are incomparable (that is, no one of the relations i j or j i holds).

Gregori-Ferrer-Cauchy nets
Later, Gregori and Ferrer [10] found a gap in the proof of Theorem 2.5 from [9] and provided a counterexample to it, based on Example 2.4 of Stoltenberg (see Proposition 10). Consider two sets A, B ∈ A with A B and let φ : I → S be given by Then the net φ is right Cauchy in the sense of (10) but not convergent in (S, d).
Indeed, for 0 < ε < 1 let k 0 ∈ N be such that 2 −k 0 < ε. Since i a, i b, i k 0 , ∀i ∈ N, k 0 a b, k 0 b a, it follows that the condition i j can hold for some i, j ∈ I, i, j k 0 , in the following cases: (a) i, j ∈ N, i, j k 0 , j < i; In the case (a), X A j X A i and In the case (b), X A j X A ∞ and again The case (c) is similar to (b).
To show that φ is not convergent let X ∈ S \ {X B ∞ }. Then b i for any i ∈ I and so that φ does not converge to X. If X = X B ∞ , then a i for any i ∈ I and Gregori and Ferrer [10] proposed a new definition of a right K-Cauchy net, for which the equivalence to sequential completeness holds.

Definition 6.
A net (x i : i ∈ I) in a quasi-metric space (X, d) is called GF-Cauchy if one of the following conditions holds: (a) for every maximal element j ∈ I the net (x i ) converges to x j ; (b) I has no maximal elements and the net (x i ) converges; (c) I has no maximal elements and the net (x i ) satisfies the condition (10).

Maximal elements and net convergence
For a better understanding of this definition we shall analyze the relations between maximal elements in a preordered set and the convergence of nets. Recall that in the definition of a directed set (I, ) the relation is supposed to be only a preorder, i.e., reflexive and transitive and not necessarily antireflexive (see [25]). Notice that some authors suppose that in the definition of a directed set is a partial order (see, e.g., [26]). For a discussion of this matter see [27], §7.12, p. 160.
Let (I, ) be a preordered set. An element j ∈ I is called: • strictly maximal if there is no i ∈ I \ {j} with j i, or, equivalently, • maximal if j i ⇒ i j, for every i ∈ I .

1.
A strictly maximal element is maximal, and if is an order, then these notions are equivalent. 2. Every maximal element j of I is a maximum for I, i.e., i j for all i ∈ I. 3. If j is a maximal element and j ∈ I satisfies j j , then j is also a maximal element. 4. (Uniqueness of the strictly maximal element) If j is a strictly maximal element, then j = j for any maximal element j of I.
Proof. 1. These assertions are obvious. 2. Indeed, suppose that j ∈ I satisfies (12). Then, for arbitrary i ∈ I, there exists i ∈ I with i j, i. But j i implies i j and so i i j. (We use the notation i j, k for i j ∧ i k.) 3. Let i ∈ I be such that j i. Then j i and, by the maximality of j, i j j .
(b) If the net (x i : i ∈ I) satisfies (13) then it satisfies (10), i.e., every strong Stoltenberg-Cauchy net is Stoltenberg-Cauchy. If is an order, then these notions are equivalent.
(a) If j, j are maximal elements of I, then x j = x j . Hence, if I has maximal elements, then there exists x ∈ X such that x j = x for every maximal element j of I, and the net (x i ) converges to x.
(b) Consequently, the net (x i ) also satisfies the conditions (a) and (c) from Definition 6, so that, GF-completeness implies completeness with respect to (13).
Proof. 1. (a) Let i, j ∈ I with i j. Since j i and i = j if i, j are comparable, the implication (14) holds. If is an order and i = j, then j i ⇒ i j and i j ⇒ i j.
(b) Since it suffices to ask that (10) and (13) hold only for distinct i, j i ε , the equivalence of these notions in the case when is an order follows.
Suppose that the net (x i ) satisfies (13). For ε > 0 choose i ε ∈ I according to (13) and let i, j i ε with i j. Taking into account (14) it follows d(x i , x j ) < ε, i.e., (x i ) satisfies (10).
Suppose now that every net satisfying (10) converges and let (x i ) be a net in X satisfying (13). Then it satisfies (10) so it converges.
2. (a) Let j, j be maximal elements of I. For ε > 0 choose i ε according to (13). By Remark 7.2, j, j i ε , j j , j j, so that d(x j , x j ) < ε and d(x j , x j ) < ε. Since these inequalities hold for every The assertions on GF-Cauchy nets follow from (a).
The following example shows that the notion of strong Stoltenberg-Cauchy net is effectively stronger that that of Stoltenberg-Cauchy net.
Example 4. Let X = R and u(x) = x + , x ∈ X be the asymmetric norm defined in Example 2. Then d u (x, y) = u(y − x) = (y − x) + , x, y ∈ X, is a quasi-metric on X. Let I = N ∪ {a, b} be the directed set considered in Example 3. Define x k = 0 for k ∈ N, x a = 1 and x b = 2. Then (x i : i ∈ I) is Stoltenberg-Cauchy but not strongly Stoltenberg-Cauchy nor GF-Cauchy.
Indeed, for ε > 0 let i ε = 1 and i, j 1 with i j. Then j ∈ N and we distinguish three possibilities: It follows that d u (x i , x j ) = 0 < ε in all cases, showing that (x i ) satisfies the condition (10), that is, it is Stoltenberg-Cauchy.
Notice that any two elements in I are comparable. Let 0 < ε < 1. Since, for every i ε ∈ I, a, b i ε and b a, but d u (x a , x b ) = (2 − 1) + = 1, it follows that (13) fails, that is, (x i ) is not strongly Stoltenberg-Cauchy.
Since a is a maximal element of I, b i ε for any i ε ∈ I, the above equality (d u (x a , x b ) = 1) shows that the net (x i ) does not converge to x a . Consequently (x i ) is not GF-Cauchy (see Definition 6).
We show now that completeness by nets with respect to (13) is equivalent to sequential right K-completeness.
Proposition 11 ([9], Theorem 2.5). A T 1 quasi-metric space (X, d) is sequentially right K-complete if and only if every net in X satisfying (13) is d-convergent, i.e., every strongly Stoltenberg-Cauchy net is convergent.

Proof.
We have only to prove that the sequential right K-completeness implies that every net in X satisfying (13) is d-convergent.
Let (x i : i ∈ I) be a net in X satisfying (13). Let i k i k−1 , k 2, be such that (13) holds for ε = 1/2 k , k ∈ N. This is possible. Indeed, take i 1 such that (13) holds for ε = 1/2. If i 2 is such that (13) holds for ε = 2 −2 , then pick i 2 ∈ I such that i 2 i 1 , i 2 . Continuing by induction one obtains the desired sequence (i k ) k∈N .
We distinguish two cases.
Let i j 0 . Then for every k, i k Since the quasi-metric space (X, d) is T 1 , it follows x i = x j 0 for all i j 0 (see Proposition 1), so that the net (x i : i ∈ I) is d-convergent to x j 0 .
Case II. ∀j ∈ I, ∀k ∈ N, ∃k k, i k j .
The inequalities d(x i k+1 , x i k ) < 2 −k , k ∈ N, imply that the sequence (x i k ) k∈N is right K-Cauchy (see Proposition 4), so it is d-convergent to some x ∈ X.
For ε > 0 choose k 0 ∈ N such that 2 −k 0 < ε and d(x, x i k ) < ε for all k k 0 . Let i ∈ I, i i k 0 . By hypothesis, there exists k k 0 such that i k i, implying i i k ∨ i k i. Since i k 0 i k , i, by the choice of i k 0 , d(x i k , x i ) < 2 −k 0 < ε in both of these cases. But then proving the convergence of the net (x i ) to x.

The proof of Proposition 11 in the case of GF-completeness
As the result in [10] is given without proof, we shall supply one. Proof. Obviously, a proof is needed only for the case (c).
Suppose that the directed set (I, ) has no maximal elements and let (x i : i ∈ I) be a net in a quasi-metric space (X, d) satisfying (10).
The proof follows the ideas of the proof of Proposition 11 with some further details. Let i k i k+1 , k ∈ N, be a sequence of indices in I such that d(x i , x j ) < 2 −k for all i, j i k with i j. We show that we can further suppose that i k+1 i k .
Indeed, the fact that I has no maximal elements implies that for every i ∈ I there exists i ∈ I such that i i and i i.
Let i 1 ∈ I be such that (10) holds for ε = 2 −1 . Take i 1 such that i 1 i 1 and i 1 i 1 . Let i 2 i 1 be such that (10) holds for ε = 2 −2 and let i 2 ∈ I satisfying i 2 i 2 and i 2 i 2 . Then i 1 i 2 and i 2 i 1 , because i 2 i 1 i 2 would contradict the choice of i 2 .
By induction one obtains a sequence (i k ) in I satisfying i k i k+1 and i k+1 i k such that (10) is satisfied with ε = 2 −k for every i k .
We shall again consider two cases.
Let i j 0 . By (15) there exists i ∈ I such that i i and i i, implying d(x i , x i ) < 2 −k for all k k 0 , that is d(x i , x i ) = 0, so that, by T 1 , x i = x i .
We also have i j 0 because i j 0 would imply i i, in contradiction to the choice of i . But then, d(x i , x j 0 ) < 2 −k for all k k 0 , so that, as above, d(x i , x i ) = 0 and x i = x j 0 .
Consequently, x i = x j 0 for every i j 0 , proving the convergence of the net (x i ) to x j 0 .
Case II. ∀j ∈ I, ∀k ∈ N, ∃k k, i k j .
The condition d(x i k , x i k+1 ) < 2 −k , k ∈ N, implies that the sequence (x i k ) k∈N is right K-Cauchy, so that there exists x ∈ X with d(x, x i k ) → 0 as k → ∞.
For ε > 0 let k 0 ∈ N be such that 2 −k 0 < ε and d(x, x i k ) < ε for all k k 0 . Let i i k 0 . By II, for j = i and k = k 0 , there exists k k 0 such that i k i. The conditions k k 0 , i k 0 i, i k 0 i k and i k i imply for all i i k 0 .

Conclusions
The present paper shows that there are big differences between the notions of completeness in metric and in quasi-metric spaces, but in spite of this, by giving appropriate definitions we can make the things to look better. The differences are even bigger in what concerns compactness in these spaces -in contrast to the metric case, in quasi-metric spaces the notions of compactness, sequential compactness and countable compactness can be different (see [2]).
Funding: This research received no external funding.