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.
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  or ). 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 ). A survey on the relations between completeness and the existence of fixed points in various circumstances is given in .
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 , who proposed a notion of Cauchy net for which completeness agrees with the sequential completeness. Later Gregori and Ferrer  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.
2. Metric and Uniform Spaces
For a mapping on a set X consider the following conditions:
The mapping d is called a pseudometric if it satisfies (M1), (M2) and (M3). A pseudometric that also satisfies (M4) is called a metric.
The open and closed balls in a pseudometric space are defined by
A filter on a set X is a nonempty family of nonempty subsets of X satisfying the conditions
It is obvious that (F2) implies
for all and .
A base of a filter is a subset of such that every contains a .
A nonempty family of nonempty subsets of X such that
generates a filter given by
A family 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 . If the relation ≤ is further antireflexive, i.e.
for all , then ≤ is called an order. A set X equipped with a preorder (order) ≤ is called a preordered (ordered) set, denoted as .
A preordered set is called directed if for every there exists with and . A net in a set X is a mapping , where is a directed set. The alternative notation , where is also used.
A uniformity on a set X is a filter on such that
The sets in are called entourages. A base for a uniformity is a base of the filter . The composition is denoted sometimes simply by Since every entourage contains the diagonal the inclusion implies
For and put
A uniformity generates a topology on X for which the family of sets is a base of neighborhoods of any point
Let be a pseudometric space. Then the pseudometric d generates a topology for which is a base of neighborhoods for every .
The pseudometric d generates also a uniform structure on X having as basis of entourages the sets
it follows that the topology agrees with the topology generated by the pseudometric d.
A sequence in is called Cauchy (or fundamental) if for every there exists such that
a condition written also as
A sequence in a uniform space is called -Cauchy (or simply Cauchy) if for every there exists such that
It is obvious that, in the case of a pseudometric space , a sequence is Cauchy with respect to the pseudometric d if and only if it is Cauchy with respect to the uniformity .
The Cauchyness of nets in pseudometric and in uniform spaces is defined by analogy with that of sequences.
A filter in a uniform space is called -Cauchy (or simply Cauchy) if for every there exists such that
A pseudometric spaceis called complete if every Cauchy sequence in X converges. A uniform spaceis called sequentially complete if every-Cauchy sequence in X converges and complete if every-Cauchy net in X converges (or, equivalently, if every-Cauchy filter in X converges).
We can define the completeness of a subset Y of a pseudometric space 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.
For a pseudometric space the following conditions are equivalent.
The metric space X is complete.
Every Cauchy net in X is convergent.
Every Cauchy filter in is convergent.
An important result in metric spaces is Cantor characterization of completeness.
(Cantor theorem).A pseudometric space 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 of nonempty closed subsets of X
If d is a metric then this intersection contains exactly one point.
The diameter of a subset Y of a pseudometric space is defined by
3. 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 .
3.1. 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 satisfying the conditions (M1) and (M3). If d satisfies further (M4) then it called a quasi-metric. The pair is called a quasi-pseudometric space, respectively a quasi-metric space (In  the term “quasi-semimetric” is used instead of “quasi-pseudometric”). Quasi-pseudometric spaces were introduced by Wilson .
The conjugate of the quasi-pseudometric d is the quasi-pseudometric The mapping is a pseudometric on X which is a metric if and only if d is a quasi-metric.
If is a quasi-pseudometric space, then for and we define the balls in X by (1)
The topology (or ) of a quasi-pseudometric space can be defined through the family of neighborhoods of an arbitrary point :
The topological notions corresponding to d will be prefixed by d- (e.g., d-closure, d-open, etc.).
The convergence of a sequence to x with respect to called d-convergence and denoted by can be characterized in the following way
As a space equipped with two topologies, and , a quasi-pseudometric space can be viewed as a bitopological space in the sense of Kelly .
The following example of quasi-metric space is an important source of counterexamples in topology (see ).
(The Sorgenfrey line).For define a quasi-metric d by if and if A basis of open -open neighborhoods of a point is formed by the family The family of intervals forms a basis of open -open neighborhoods of Obviously, the topologies and are Hausdorff and for so that is the discrete topology of
Asymmetric normed spaces
Let X be a real vector space. A mapping is called an asymmetric seminorm on X if
for all and
for all then p is called an asymmetric norm.
To an asymmetric seminorm p one associates a quasi-pseudometric 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 (see ).
The following asymmetric norm on is essential in the study of asymmetric normed spaces (see ).
On the field of real numbers consider the asymmetric norm Then, for and The topology generated by u is called the upper topology of while the topology generated by is called the lower topology of A basis of open -neighborhoods of a point is formed of the intervals A basis of open -neighborhoods is formed of the intervals
In this space the addition is continuous from to , but the multiplication is discontinuous at every point
The multiplication is continuous from to but discontinuous at every point to , when is equipped with the topology generated by and with .
The following topological properties are true for quasi-pseudometric spaces.
(see ).If is a quasi-pseudometric space, then the following hold.
The ball is d-open and the ball is -closed. The ball need not be d-closed.
The topology d is if and only if d is a quasi-metric.
The topology d is if and only if for all in X.
For every fixed the mapping is d-upper semi-continuous and -lower semi-continuous.
For every fixed the mapping is d-lower semi-continuous and -upper semi-continuous.
Recall that a topological space is called:
if, for every pair of distinct points in X, at least one of them has a neighborhood not containing the other;
if, for every pair of distinct points in X, each of them has a neighborhood not containing the other;
(or Hausdorff) if every two distinct points in X admit disjoint neighborhoods;
regular if, for every point and closed set A not containing x, there exist the disjoint open sets such that and
It is known that the topology of a pseudometric space is Hausdorff (or ) 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 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 one -limit (see ).
In the case of an asymmetric seminormed space there exists a characterization in terms of the asymmetric seminorm (see , Proposition 1.1.40).
3.2. 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 in satisfying the conditions (U1) and (U2). The sets in are called entourages and the pair is called a quasi-uniform space, as in the case of uniform spaces.
As uniformities, a quasi-uniformity generates a topology on X in a similar way: the sets
form a base of neighborhoods of any point
The topology is if and only if is a partial order on X, and if and only if .
The family of sets
is another quasi-uniformity on X called the quasi-uniformity conjugate to . Also is a subbase of a uniformity on X, called the associated uniformity to the quasi-uniformity . It is the coarsest uniformity on X finer than both and A basis for is formed by the sets
If is a quasi-pseudometric space, then
is a basis for a quasi-uniformity on The family
generates the same quasi-uniformity. Since and , it follows that the topologies generated by the quasi-pseudometric d and by the quasi-uniformity agree, i.e.,
In this case
Quasi-uniform spaces were studied by Nachbin starting with 1948 (see ) as a generalization of uniform spaces introduced by Weil . For further developments see [16,17].
4. 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.  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 . 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.
-Cauchy ⇒ left K-Cauchy weakly left K-Cauchy left d-Cauchy.
The same implications hold for the corresponding right notions. No one of the above implications is reversible.
A sequence is left Cauchy (in some sense) with respect to d if and only if it is right Cauchy (in the same sense) with respect to
A sequence is -Cauchy if and only if it is both left and right d-K-Cauchy.
A d-convergent sequence is left d-Cauchy and a -convergent sequence is right d-Cauchy. For the other notions, a convergent sequence need not be Cauchy.
If each convergent sequence in a regular quasi-metric space admits a left K-Cauchy subsequence, then X is metrizable ().
We also mention the following simple properties of Cauchy sequences.
([19,20]).Let be a left or right K-Cauchy sequence in a quasi-pseudometric space
If has a subsequence which is d-convergent to then is d-convergent to
If has a subsequence which is -convergent to then is -convergent to
If has a subsequence which is -convergent to x, then is -convergent to x.
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 -convergent. Due to the equivalence d-left Cauchy ⇔-right Cauchy one obtains nothing new by asking that a d-left Cauchy sequence is -convergent. For instance, the -convergence of any left d-K-Cauchy sequence is equivalent to the right K-completeness of the space
sequentially d-complete if every -Cauchy sequence is d-convergent;
sequentially left d-complete if every left d-Cauchy sequence is d-convergent;
sequentially weakly left (right) K-complete if every weakly left (right) K-Cauchy sequence is d-convergent;
sequentially left (right) K-complete if every left (right) K-Cauchy sequence is d-convergent;
sequentially left (right) Smyth complete if every left (right) K-Cauchy sequence is -convergent;
bicomplete if the associated pseudometric space is complete, i.e., every -Cauchy sequence is -convergent. A bicomplete asymmetric normed space 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).
(a) These notions of completeness are related in the following way:
sequentially d-complete sequentially weakly left K-complete sequentially left K-complete sequentially left d-complete.
The same implications hold for the corresponding notions of right completeness.
(b) sequentially left or right Smyth completeness implies bicompleteness.
No one of the above implication is reversible (see ), excepting that between weakly left and left K-sequential completeness, as it was surprisingly shown by Romaguera .
(, Proposition 1).A quasi-pseudometric space is sequentially weakly left K-complete if and only if it is sequentially left K-complete.
A series in an asymmetric seminormed space is called convergent if there exists such that (i.e., ). The series is called absolutely convergent if 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.
(, Proposition 1.2.6).Let be a quasi-pseudometric space.
If a sequence in X satisfies then it is left (right) d-K-Cauchy.
The quasi-pseudometric space is sequentially left (right) d-K-complete if and only if every sequence in X satisfying (resp. is d-convergent.
An asymmetric seminormed space 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 is defined by
It is clear that, as defined, the diameter is, in fact, the diameter with respect to the associated pseudometric . Recall that a quasi-pseudometric space is called sequentially d-complete if every -Cauchy sequence is d-convergent (see Definition 3).
(, Theorem 10).A quasi-pseudometric space is sequentially d-complete if and only if each decreasing sequence of nonempty closed sets with as has nonempty intersection, which is a singleton if d is a quasi-metric.
The following characterization of right K-completeness was obtained in , using a different terminology.
A quasi-pseudometric space is sequentially right K-complete if and only if any decreasing sequence of closed -balls
has nonempty intersection.
If the topology d is Hausdorff, then contains exactly one element.
5. 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 , they can be different for others.
5.1. 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 in a quasi-pseudometric space 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).
(, Prop. 1).For a quasi-metric space the following are equivalent.
Every left d-K-Cauchy sequence is -convergent.
Every left d-K-Cauchy net is -convergent.
A quasi-uniform space is called bicomplete if 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.
Let be a quasi-uniform space.
A filter on is called:
left (right) -Cauchy if for every there exists such that (respectively );
left (right) -K- Cauchy if for every there exists such that (resp. ) for all .
A net in is called:
left -Cauchy (right -Cauchy) if for every there exists and such that (respectively for all ;
left -K-Cauchy (right -K-Cauchy) if
The notions of left and right -K-Cauchy filter were defined by Romaguera in .
so that a filter is right -K-Cauchy if and only if it is left -K-Cauchy. A similar remark applies to -nets.
A quasi-uniform space is called:
left -complete by filters (left K-complete by filters) if every left -Cauchy (respectively, left -K-Cauchy) filter in X is -convergent;
left -complete by nets (left -K-complete by nets) if every left -Cauchy (respectively, left -K-Cauchy) net in X is -convergent;
Smyth left -K-complete by nets if every left K-Cauchy net in X is -convergent.
The notions of right completeness are defined similarly, by asking the -convergence of the corresponding right Cauchy filter (or net) with respect to the topology (or with respect to 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  proved a similar result for the left K-completeness in quasi-pseudometric spaces.
In the case of a quasi-pseudometric space the considered notions take the following form.
A filter in a quasi-pseudometric space is called left K-Cauchy if it left -K-Cauchy. This is equivalent to the fact that for every there exists such that
Also a net is called left K-Cauchy if it is left -K-Cauchy or, equivalently, for every there exists such that
().For a quasi-pseudometric space the following are equivalent.
The space is sequentially left K-complete.
Every left K-Cauchy filter in X is d-convergent.
Every left K-Cauchy net in X is d-convergent.
In the case of left -completeness this equivalence does not hold in general.
(Künzi ).A Hausdorff quasi-metric space is sequentially left d-complete if and only if the associated quasi-uniform space is left -complete by filters.
5.2. 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  (Example 2.4), a result similar to Proposition 7 does not hold for right K-completeness: there exists a sequentially right K-complete quasi-metric space which is not right K-complete by nets. Actually, Stoltenberg  proved that the equivalence holds for a more general definition of a right K-Cauchy net, but Gregori and Ferrer  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 is called if for all -- implies the existence of two disjoint d-open sets such that and
().Let be a quasi-pseudometric space. The following are true.
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.
If the quasi-pseudometric space is then X is right K-complete by filters if and only if it is sequentially right K-complete.
As we have mentioned, Stoltenberg  (Example 2.4), gave an example of a sequentially right K-complete quasi-metric space which is not right K-complete by nets, which we shall present now.
Denote by the family of all countable subsets of the interval . For let
Put and define by
is a sequentially right K-complete quasi-metric space which is not right K-complete by nets.
The proof that d is a quasi-metric on is straightforward.
I. is sequentially rightK-complete.
Let be a right K-Cauchy sequence in . Then there exists such that
it follows for all For there exists such that
which means that
This shows that
Let and Then , so that
which shows that the sequence is d-convergent to X.
II. The quasi-metric space is not right K-complete by nets.
Let ordered by
for , is directed and the mapping defined by is a net in
Let us show first that the net is right K-Cauchy. For choose such that For some belongs to and
showing that the net is right K-Cauchy.
Let be an arbitrary element in We show that for every there exists with such that which will imply that the net is not d-convergent to X.
Since C is a countable set, there exists For an arbitrary let . Then , and , so that, by the definition of the metric d, □
Stoltenberg  also considered a more general definition of a right K-Cauchy net as a net satisfying the condition: for every there exists 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 means that are incomparable (that is, no one of the relations or holds).
Later, Gregori and Ferrer  found a gap in the proof of Theorem 2.5 from  and provided a counterexample to it, based on Example 2.4 of Stoltenberg (see Proposition 10).
().Let , be as in the preamble to Proposition 10 and , where the set is considered with the usual order and are two distinct elements not belonging to with
Consider two sets with and let be given by
Then the net ϕ is right Cauchy in the sense of (10) but not convergent in .
Indeed, for let be such that
it follows that the condition can hold for some , in the following cases:
In the case (a), and
In the case (b), and again
The case (c) is similar to (b).
To show that is not convergent let Then for any and
so that does not converge to X. If , then for any and
Gregori and Ferrer  proposed a new definition of a right K-Cauchy net, for which the equivalence to sequential completeness holds.
A net in a quasi-metric space is called -Cauchy if one of the following conditions holds:
for every maximal element the net converges to ;
I has no maximal elements and the net converges;
I has no maximal elements and the net 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 the relation is supposed to be only a preorder, i.e., reflexive and transitive and not necessarily antireflexive (see ). Notice that some authors suppose that in the definition of a directed set is a partial order (see, e.g., ). For a discussion of this matter see , §7.12, p. 160.
Let be a preordered set. An element is called:
strictly maximal if there is no with or, equivalently,
Let be a preordered set.
A strictly maximal element is maximal, and if ⩽ is an order, then these notions are equivalent.
Every maximal element j of I is a maximum for I, i.e., for all .
If j is a maximal element and satisfies , then is also a maximal element.
(Uniqueness of the strictly maximal element) If j is a strictly maximal element, then for any maximal element of I.
1. These assertions are obvious.
2. Indeed, suppose that satisfies (12). Then, for arbitrary , there exists with But implies and so (We use the notation for .)
3. Let be such that . Then and, by the maximality of j, .
4. If j is strictly maximal and is a maximal element of I, then, by 2, so that, by (12) applied to , and so, by (11) applied to j, □
We present now some remarks on maximal elements and net convergence.
Let be a quasi-metric space, a directed sets and a net in X.
If has a strictly maximal element j, then the net is convergent to .
If the net converges to , then for every maximal element j of I. If the topology is then, further,
If the net converges to and to , where are maximal elements of I, then .
If I has maximal elements and, for some , for every maximal element j, then the net converges to x.
1. For an arbitrary take Then implies , so that
For every there exists such that for all By Remark 7.2, for every maximal j, so that for all , implying .
If the topology is , then, by Proposition 1.2, .
By (a), and , so that .
Let be such that for every maximal element j of I and let j be a fixed maximal element of I. For any put . Then, by Remark 7.3, any such that is also a maximal element of I, so that and
Let us say that a quasi-metric space is GF-complete if every GF-Cauchy net (i.e., satisfying the conditions (a), (b), (c) from Definition 6) is convergent. Remark that, with this definition, condition (b) becomes tautological and so superfluous, so it suffices to ask that every net satisfying (a) and (c) be convergent.
By Remarks 7.1 and 8.1, (a) always holds if ⩽ is an order, so that, in this case, a net satisfying condition (c) is a GF-Cauchy net and so GF-completeness agrees with that given by Stoltenberg.
Strongly Stoltenberg-Cauchy nets
In order to avoid the shortcomings of the preorder relation, as, for instance, those put in evidence by Example 3, we propose the following definition.
A net in a quasi-metric space is called strongly Stoltenberg-Cauchy if for every there exists such that, for all ,
We present now some remarks on the relations of this notion with the other notions of Cauchy net (Stoltenberg and GF), as well as the relations between the corresponding completeness notions. It is obvious that in the case of a sequence each of these three notions agrees with the right K-Cauchyness of .
Let be a net in a quasi-metric space .
1. (a) We have
for all . If ⩽ is an order, then the reverse implication also holds.
(b) If the net 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.
Hence, net completeness with respect to (10) (i.e. Stoltenberg completeness) implies net completeness with respect to (13);
(a) If are maximal elements of I, then . Hence, if I has maximal elements, then there exists such that for every maximal element j of I, and the net converges to
(b) Consequently, the net also satisfies the conditions (a) and (c) from Definition 6, so that, GF-completeness implies completeness with respect to (13).
1. (a) Let with . Since and if are comparable, the implication (14) holds. If ⩽ is an order and , then and .
(b) Since it suffices to ask that (10) and (13) hold only for distinct , the equivalence of these notions in the case when ⩽ is an order follows.
Suppose that the net satisfies (13). For choose according to (13) and let with Taking into account (14) it follows , i.e., satisfies (10).
Suppose now that every net satisfying (10) converges and let be a net in X satisfying (13). Then it satisfies (10) so it converges.
2. (a) Let be maximal elements of I. For choose according to (13). By Remark 7.2, , so that and . Since these inequalities hold for every , it follows and so . The convergence of the net follows from Remark 8.2.(c).
(b) 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.
Let and be the asymmetric norm defined in Example 2. Then is a quasi-metric on X. Let be the directed set considered in Example 3. Define for and . Then is Stoltenberg-Cauchy but not strongly Stoltenberg-Cauchy nor GF-Cauchy.
Indeed, for let and with . Then and we distinguish three possibilities:
and when ;
It follows that in all cases, showing that satisfies the condition (10), that is, it is Stoltenberg-Cauchy.
Notice that any two elements in I are comparable. Let Since, for every and , but , it follows that (13) fails, that is, is not strongly Stoltenberg-Cauchy.
Since a is a maximal element of I, for any , the above equality () shows that the net does not converge to . Consequently 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.
(, Theorem 2.5).A quasi-metric space 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.
We have only to prove that the sequential right K-completeness implies that every net in X satisfying (13) is d-convergent.
Let be a net in X satisfying (13). Let be such that (13) holds for
This is possible. Indeed, take such that (13) holds for If is such that (13) holds for , then pick such that Continuing by induction one obtains the desired sequence
We distinguish two cases.
Let Then for every k, implies so that Since the quasi-metric space is it follows for all (see Proposition 1), so that the net is d-convergent to
The inequalities imply that the sequence is right K-Cauchy (see Proposition 4), so it is d-convergent to some
For choose such that and for all
Let By hypothesis, there exists such that implying Since , by the choice of , in both of these cases. But then
proving the convergence of the net to □
The proof of Proposition 11 in the case of GF-completeness
As the result in  is given without proof, we shall supply one.
A quasi-metric space is right K-sequentially complete if and only if every net satisfying the conditions (a) and (c) from Definition 6 is convergent.
Obviously, a proof is needed only for the case (c).
Suppose that the directed set has no maximal elements and let be a net in a quasi-metric space satisfying (10).
The proof follows the ideas of the proof of Proposition 11 with some further details. Let be a sequence of indices in I such that for all with . We show that we can further suppose that .
Indeed, the fact that I has no maximal elements implies that for every there exists such that
Let be such that (10) holds for Take such that and . Let be such that (10) holds for and let satisfying and . Then and , because would contradict the choice of
By induction one obtains a sequence in I satisfying and such that (10) is satisfied with for every .
We shall again consider two cases.
Let . By (15) there exists such that and implying for all , that is , so that, by , .
We also have because would imply , in contradiction to the choice of . But then, for all , so that, as above, and .
Consequently, for every , proving the convergence of the net to .
The condition implies that the sequence is right K-Cauchy, so that there exists with as .
For let be such that and for all .
Let . By II, for and , there exists such that . The conditions and imply
for all . □
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 ).
This research received no external funding.
The author expresses his deep gratitude to reviewers for the careful reading of the manuscript and for the suggestions that led to improvements both in presentation and in contents.
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