#### 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 $({x}_{i}:i\in I)$ 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).

**Proposition** **6** ([

6], Prop. 1)

**.** For a quasi-metric space $(X,d)$ the following are equivalent.- 1.
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 $(X,\mathcal{U})$ is called bicomplete if $(X,{\mathcal{U}}^{s})$ 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 $(X,\mathcal{U})$ be a quasi-uniform space.

A filter $\mathcal{F}$ on $(X,\mathcal{U})$ is called:

left (right) $\mathcal{U}$-Cauchy if for every $U\in \mathcal{U}$ there exists $x\in X$ such that $U\left(x\right)\in \mathcal{F}$ (respectively ${U}^{-1}\left(x\right)\in \mathcal{F}$);

left (right) $\mathcal{U}$-K- Cauchy if for every $U\in \mathcal{U}$ there exists $F\in \mathcal{F}$ such that $U\left(x\right)\in \mathcal{F}$ (resp. ${U}^{-1}\left(x\right)\in \mathcal{F}$) for all $x\in F$.

A net $({x}_{i}:i\in I)$ in $(X,\mathcal{U})$ is called:

left $\mathcal{U}$-Cauchy (right $\mathcal{U}$-Cauchy) if for every $U\in \mathcal{U}$ there exists $x\in X$ and ${i}_{0}\in I$ such that $(x,{x}_{i})\in U$ (respectively $({x}_{i},x)\in U)$ for all $i\u2a7e{i}_{0}$;

left $\mathcal{U}$-K-Cauchy (right $\mathcal{U}$-K-Cauchy) if

The notions of left and right

$\mathcal{U}$-

K-Cauchy filter were defined by Romaguera in [

21].

Observe that

so that a filter is right

$\mathcal{U}$-

K-Cauchy if and only if it is left

${\mathcal{U}}^{-1}$-

K-Cauchy. A similar remark applies to

$\mathcal{U}$-nets.

**Definition** **5.** A quasi-uniform space $(X,\mathcal{U})$ is called:

left $\mathcal{U}$-complete by filters (left K-complete by filters) if every left $\mathcal{U}$-Cauchy (respectively, left $\mathcal{U}$-K-Cauchy) filter in X is $\tau \left(\mathcal{U}\right)$-convergent;

left $\mathcal{U}$-complete by nets (left $\mathcal{U}$-K-complete by nets) if every left $\mathcal{U}$-Cauchy (respectively, left $\mathcal{U}$-K-Cauchy) net in X is $\tau \left(\mathcal{U}\right)$-convergent;

Smyth left $\mathcal{U}$-K-complete by nets if every left K-Cauchy net in X is ${\mathcal{U}}^{s}$-convergent.

The notions of right completeness are defined similarly, by asking the $\tau \left(\mathcal{U}\right)$-convergence of the corresponding right Cauchy filter (or net) with respect to the topology $\tau \left(\mathcal{U}\right)$ (or with respect to $\tau \left({\mathcal{U}}^{s}\right)$ 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 $\mathcal{F}$ in a quasi-pseudometric space $(X,d)$ is called left

K-Cauchy

if it left ${\mathcal{U}}_{d}$-K-Cauchy. This is equivalent to the fact that for every $\epsilon >0$ there exists ${F}_{\epsilon}\in \mathcal{F}$ such that Also a net $({x}_{i}:i\in I)$ is called left

K-Cauchy

if it is left ${\mathcal{U}}_{d}$-K-Cauchy or, equivalently, for every $\epsilon >0$ there exists ${i}_{0}\in I$ 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.

- 2.
Every left K-Cauchy filter in X is d-convergent.

- 3.
Every left K-Cauchy net in X is d-convergent.

In the case of left ${\mathcal{U}}_{d}$-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 $(X,{\mathcal{U}}_{d})$ is left ${\mathcal{U}}_{d}$-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 [

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 $x,y\in X,\phantom{\rule{4pt}{0ex}}d$-$cl\left\{x\right\}\ne d$-$cl\left\{y\right\}$ implies the existence of two disjoint d-open sets $U,V$ such that $x\in U$ and $y\in V.$

**Proposition** **9** ([

24])

**.** Let $(X,d)$ be a quasi-pseudometric space. The following are true.- 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

$\mathcal{A}$ the family of all countable subsets of the interval

$[0,\frac{1}{3}]$. For

$A\in \mathcal{A}$ let

Put

$\mathcal{S}=\left\{{X}_{k}^{A}:A\in \mathcal{A},\phantom{\rule{0.166667em}{0ex}}k\in \mathbb{N}\cup \{\infty \}\right\}$ and define

$d:\mathcal{S}\times \mathcal{S}\to [0,\infty )$ by

**Proposition** **10.** $(\mathcal{S},d)$ 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 $\mathcal{S}$ is straightforward.

I. $(\mathcal{S},d)$is sequentially rightK-complete.

Let

${\left({X}_{n}\right)}_{n\in \mathbb{N}}$ be a right

K-Cauchy sequence in

$\mathcal{S}$. Then there exists

${n}_{0}\in \mathbb{N}$ such that

For

$i\in {\mathbb{N}}_{0}=\mathbb{N}\cup \left\{0\right\}$ let

Since

it follows

${k}_{i}\in \mathbb{N}$ for all

$i\in {\mathbb{N}}_{0}.$ For

$0<\epsilon <1$ there exists

${i}_{0}\in \mathbb{N}$ such that

which means that

This shows that ${lim}_{i\to \infty}{k}_{i}=\infty .$

Let

$A=\bigcup \left\{{A}_{i}:i\in {\mathbb{N}}_{0}\right\}$ and

$X={X}_{\infty}^{A}.$ Then

${X}_{{k}_{i}}^{{A}_{i}}\u2acb{X}_{\infty}^{A}$, so that

which shows that the sequence

$\left({X}_{n}\right)$ is

d-convergent to

X.

II. The quasi-metric space $(\mathcal{S},d)$ is not right K-complete by nets.

Let

${\mathcal{S}}_{0}=\{{X}_{k}^{A}:A\in \mathcal{A},\phantom{\rule{0.166667em}{0ex}}k\in \mathbb{N}\}$ ordered by

We have

for

${X}_{i}^{A},{X}_{j}^{B}\in {\mathcal{S}}_{0}$,

$({\mathcal{S}}_{0},\u2a7d)$ is directed and the mapping

$\varphi :{\mathcal{S}}_{0}\to \mathcal{S}$ defined by

$\varphi \left(X\right)=X,\phantom{\rule{0.166667em}{0ex}}X\in {\mathcal{S}}_{0},$ is a net in

$\mathcal{S}.$Let us show first that the net

$\varphi $ is right

K-Cauchy. For

$\epsilon >0$ choose

$k\in \mathbb{N}$ such that

${2}^{-k}<\epsilon .$ For some

$C\in \mathcal{A},\phantom{\rule{0.166667em}{0ex}}{X}_{k}^{C}$ belongs to

${\mathcal{S}}_{0}$ and

for all

showing that the net

$\varphi $ is right

K-Cauchy.

Let $X={X}_{k}^{C}$ be an arbitrary element in $\mathcal{S}.$ We show that for every ${X}_{i}^{A}\in {\mathcal{S}}_{0}$ there exists ${X}_{j}^{B}\in {\mathcal{S}}_{0}$ with ${X}_{i}^{A}\u2a7d{X}_{j}^{B}$ such that $d(X,{X}_{j}^{B})=1,$ which will imply that the net $\varphi $ is not d-convergent to X.

Since C is a countable set, there exists ${x}_{0}\in [0,\frac{1}{3}]\backslash C.$ For an arbitrary ${X}_{i}^{A}\in {\mathcal{S}}_{0}$ let $B=A\cup \left\{{x}_{0}\right\}$. Then ${X}_{i}^{B}\in {\mathcal{S}}_{0}$, ${X}_{i}^{A}\u2a7d{X}_{i}^{B}$ and ${X}_{k}^{C}\u2288{X}_{i}^{B}$, so that, by the definition of the metric d, $d({X}_{k}^{C},{X}_{i}^{B})=1.$ □

**Stoltenberg-Cauchy nets**

Stoltenberg [

9] also considered a more general definition of a right

K-Cauchy net as a net

$({x}_{i}:i\in I)$ satisfying the condition: for every

$\epsilon >0$ there exists

${i}_{\epsilon}\in I$ 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

$i\nsim j$ means that

$i,j$ are incomparable (that is, no one of the relations

$i\u2a7dj$ or

$j\u2a7di$ 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).

**Example** **3** ([

10])

**.** Let $\mathcal{A}$, $(\mathcal{S},d)$ be as in the preamble to Proposition 10 and $I=\mathbb{N}\cup \{a,b\}$, where the set $\mathbb{N}$ is considered with the usual order and $a,b$ are two distinct elements not belonging to $\mathbb{N}$ withConsider two sets $A,B\in \mathcal{A}$ with $A\u2acbB$ and let $\varphi :I\to \mathcal{S}$ be given by Then the net ϕ is right Cauchy in the sense of (10) but not convergent in $(\mathcal{S},d)$. Indeed, for $0<\epsilon <1$ let ${k}_{0}\in \mathbb{N}$ be such that ${2}^{-{k}_{0}}<\epsilon .$

Since

it follows that the condition

$i\nleqq j$ can hold for some

$i,j\in I,\phantom{\rule{0.166667em}{0ex}}i,j\u2a7e{k}_{0}$, in the following cases:

In the case (a),

${X}_{j}^{A}\u2acb{X}_{i}^{A}$ and

In the case (b),

${X}_{j}^{A}\u2acb{X}_{\infty}^{A}$ and again

The case (c) is similar to (b).

To show that

$\varphi $ is not convergent let

$X\in \mathcal{S}\backslash \left\{{X}_{\infty}^{B}\right\}.$ Then

$b\u2a7ei$ for any

$i\in I$ and

so that

$\varphi $ does not converge to

X. If

$X={X}_{\infty}^{B}$, then

$a\u2a7ei$ for any

$i\in I$ 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 $({x}_{i}:i\in I)$ in a quasi-metric space $(X,d)$ is called $GF$-Cauchy if one of the following conditions holds:

- (a)
for every maximal element $j\in I$ the net $\left({x}_{i}\right)$ converges to ${x}_{j}$;

- (b)
I has no maximal elements and the net $\left({x}_{i}\right)$ converges;

- (c)
I has no maximal elements and the net $\left({x}_{i}\right)$ 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

$(I,\u2a7d)$ the relation

$\phantom{\rule{0.166667em}{0ex}}\u2a7d\phantom{\rule{0.166667em}{0ex}}$ 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

$\phantom{\rule{0.166667em}{0ex}}\u2a7d\phantom{\rule{0.166667em}{0ex}}$ is a partial order (see, e.g., [

26]). For a discussion of this matter see [

27], §7.12, p. 160.

Let $(I,\u2a7d)$ be a preordered set. An element $j\in I$ is called:

strictly maximal if there is no

$i\in I\backslash \left\{j\right\}$ with

$j\u2a7di,$ or, equivalently,

**Remark** **7.** Let $(I,\u2a7d)$ be a preordered set.

- 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., $i\u2a7dj$ for all $i\in I$.

- 3.
If j is a maximal element and ${j}^{\prime}\in I$ satisfies $j\u2a7d{j}^{\prime}$, then ${j}^{\prime}$ is also a maximal element.

- 4.
(Uniqueness of the strictly maximal element) If j is a strictly maximal element, then ${j}^{\prime}=j$ for any maximal element ${j}^{\prime}$ of I.

**Proof.** 1. These assertions are obvious.

2. Indeed, suppose that

$j\in I$ satisfies (

12). Then, for arbitrary

$i\in I$, there exists

${i}^{\prime}\in I$ with

${i}^{\prime}\u2a7ej,i.$ But

$j\u2a7d{i}^{\prime}$ implies

${i}^{\prime}\u2a7dj$ and so

$i\u2a7d{i}^{\prime}\u2a7dj.$ (We use the notation

$i\u2a7ej,k$ for

$i\u2a7ej\wedge i\u2a7ek$.)

3. Let $i\in I$ be such that ${j}^{\prime}\u2a7di$. Then $j\u2a7di$ and, by the maximality of j, $i\u2a7dj\u2a7d{j}^{\prime}$.

4. If

j is strictly maximal and

${j}^{\prime}$ is a maximal element of

I, then, by 2,

${j}^{\prime}\le j$ so that, by (

12) applied to

${j}^{\prime}$,

$j\le {j}^{\prime}$ and so, by (

11) applied to

j,

${j}^{\prime}=j.$ □

We present now some remarks on maximal elements and net convergence.

**Remark** **8.** Let $(X,d)$ be a quasi-metric space, $(I,\u2a7d)$ a directed sets and $({x}_{i}:i\in I)$ a net in X.

- 1.
If $(I,\u2a7d)$ has a strictly maximal element j, then the net $\left({x}_{i}\right)$ is convergent to ${x}_{j}$.

- 2.
- (a)
If the net $\left({x}_{i}\right)$ converges to $x\in X$, then $d(x,{x}_{j})=0$ for every maximal element j of I. If the topology ${\tau}_{d}$ is ${T}_{1}$ then, further, ${x}_{j}=x.$

- (b)
If the net $\left({x}_{i}\right)$ converges to ${x}_{j}$ and to ${x}_{{j}^{\prime}}$, where $j,{j}^{\prime}$ are maximal elements of I, then ${x}_{j}={x}_{{j}^{\prime}}$.

- (c)
If I has maximal elements and, for some $x\in X$, ${x}_{j}=x$ for every maximal element j, then the net $\left({x}_{i}\right)$ converges to x.

**Proof.** 1. For an arbitrary

$\epsilon >0$ take

${i}_{\epsilon}=j.$ Then

$i\u2a7ej$ implies

$i=j$, so that

- 2.
- (a)
For every $\epsilon >0$ there exists ${i}_{\epsilon}\in I$ such that $d(x,{x}_{i})<\epsilon $ for all $i\u2a7e{i}_{\epsilon}.$ By Remark 7.2, $j\u2a7e{i}_{\epsilon}$ for every maximal j, so that $d(x,{x}_{j})<\epsilon $ for all $\epsilon >0$, implying $d(x,{x}_{j})=0$.

If the topology ${\tau}_{d}$ is ${T}_{1}$, then, by Proposition 1.2, ${x}_{j}=x$.

- (b)
By (a), $d({x}_{j},{x}_{{j}^{\prime}})=0$ and $d({x}_{{j}^{\prime}},{x}_{j})=0$, so that ${x}_{j}={x}_{{j}^{\prime}}$.

- (c)
Let $x\in X$ be such that ${x}_{j}=x$ for every maximal element j of I and let j be a fixed maximal element of I. For any $\epsilon >0$ put ${i}_{\epsilon}=j$. Then, by Remark 7.3, any $i\in I$ such that $i\u2a7ej$ is also a maximal element of I, so that ${x}_{i}=x$ and $d(x,{x}_{i})=0<\epsilon .$

□

Let us say that a quasi-metric space $(X,d)$ 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.

**Definition** **7.** A net $({x}_{i}:i\in I)$ in a quasi-metric space $(X,d)$ is called strongly Stoltenberg-Cauchy

if for every $\epsilon >0$ there exists ${i}_{\epsilon}\in I$ such that, for all $i,j\u2a7e{i}_{\epsilon}$, 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 ${\left({x}_{k}\right)}_{k\in \mathbb{N}}$ each of these three notions agrees with the right K-Cauchyness of $\left({x}_{k}\right)$.

**Remark** **9.** Let $({x}_{i}:i\in I)$ be a net in a quasi-metric space $(X,d)$.

1. (a) We havefor all $i,j\in I$. If ⩽ is an order, then the reverse implication also holds. (b) If the net $({x}_{i}:i\in I)$ 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. Hence, net completeness with respect to (10) (i.e. Stoltenberg completeness) implies net completeness with respect to (13); 2. Suppose that the net $({x}_{i}:i\in I)$ satisfies (13). (a) If $j,{j}^{\prime}$ are maximal elements of I, then ${x}_{j}={x}_{{j}^{\prime}}$. Hence, if I has maximal elements, then there exists $x\in X$ such that ${x}_{j}=x$ for every maximal element j of I, and the net $\left({x}_{i}\right)$ converges to $x.$

(b) Consequently, the net $\left({x}_{i}\right)$ also satisfies the conditions (a) and (c) from Definition 6, so that, GF-completeness implies completeness with respect to (13). **Proof.** 1. (a) Let

$i,j\in I$ with

$i\nleqq j$. Since

$j\u2a7di$ and

$i\ne j$ if

$i,j$ are comparable, the implication (

14) holds. If ⩽ is an order and

$i\ne j$, then

$j\u2a7di\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}i\nleqq j$ and

$i\nsim j\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}i\nleqq j$.

(b) Since it suffices to ask that (

10) and (

13) hold only for distinct

$i,j\u2a7e{i}_{\epsilon}$, the equivalence of these notions in the case when ⩽ is an order follows.

Suppose that the net

$\left({x}_{i}\right)$ satisfies (

13). For

$\epsilon >0$ choose

${i}_{\epsilon}\in I$ according to (

13) and let

$i,j\u2a7e{i}_{\epsilon}$ with

$i\nleqq j.$ Taking into account (

14) it follows

$d({x}_{i},{x}_{j})<\epsilon $, i.e.,

$\left({x}_{i}\right)$ satisfies (

10).

Suppose now that every net satisfying (

10) converges and let

$\left({x}_{i}\right)$ be a net in

X satisfying (

13). Then it satisfies (

10) so it converges.

2. (a) Let

$j,{j}^{\prime}$ be maximal elements of

I. For

$\epsilon >0$ choose

${i}_{\epsilon}$ according to (

13). By Remark 7.2,

$j,{j}^{\prime}\u2a7e{i}_{\epsilon},\phantom{\rule{0.166667em}{0ex}}j\u2a7d{j}^{\prime},\phantom{\rule{4pt}{0ex}}{j}^{\prime}\u2a7dj$, so that

$d({x}_{{j}^{\prime}},{x}_{j})<\epsilon $ and

$d({x}_{j},{x}_{{j}^{\prime}})<\epsilon $. Since these inequalities hold for every

$\epsilon >0$, it follows

$d({x}_{{j}^{\prime}},{x}_{j})=0=d({x}_{j},{x}_{{j}^{\prime}})$ and so

${x}_{j}={x}_{{j}^{\prime}}$. The convergence of the net

$\left({x}_{i}\right)$ 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.

**Example** **4.** Let $X=\mathbb{R}$ and $u\left(x\right)={x}^{+},\phantom{\rule{0.166667em}{0ex}}x\in X$ be the asymmetric norm defined in Example 2. Then ${d}_{u}(x,y)=u(y-x)={(y-x)}^{+},\phantom{\rule{0.166667em}{0ex}}x,y\in X,$ is a quasi-metric on X. Let $I=\mathbb{N}\cup \{a,b\}$ be the directed set considered in Example 3. Define ${x}_{k}=0$ for $k\in \mathbb{N},\phantom{\rule{0.166667em}{0ex}}{x}_{a}=1$ and ${x}_{b}=2$. Then $({x}_{i}:i\in I)$ is Stoltenberg-Cauchy but not strongly Stoltenberg-Cauchy nor GF-Cauchy.

Indeed, for $\epsilon >0$ let ${i}_{\epsilon}=1$ and $i,j\u2a7e1$ with $i\nleqq j$. Then $j\in \mathbb{N}$ and we distinguish three possibilities:

$i\in \mathbb{N}$ and $j<i\phantom{\rule{0.166667em}{0ex}},$ when ${d}_{u}({x}_{i},{x}_{j})={({x}_{j}-{x}_{i})}^{+}=0$;

$i=a$ and ${d}_{u}({x}_{a},{x}_{j})={({x}_{j}-{x}_{a})}^{+}={(0-1)}^{+}=0$;

$i=b$ and ${d}_{u}({x}_{b},{x}_{j})={({x}_{j}-{x}_{b})}^{+}={(0-2)}^{+}=0$.

It follows that

${d}_{u}({x}_{i},{x}_{j})=0<\epsilon $ in all cases, showing that

$\left({x}_{i}\right)$ satisfies the condition (

10), that is, it is Stoltenberg-Cauchy.

Notice that any two elements in

I are comparable. Let

$0<\epsilon <1.$ Since, for every

${i}_{\epsilon}\in I,\phantom{\rule{0.166667em}{0ex}}a,b\u2a7e{i}_{\epsilon}$ and

$\phantom{\rule{4pt}{0ex}}b\u2a7da$, but

${d}_{u}({x}_{a},{x}_{b})={(2-1)}^{+}=1$, it follows that (

13) fails, that is,

$\left({x}_{i}\right)$ is not strongly Stoltenberg-Cauchy.

Since a is a maximal element of I, $b\u2a7e{i}_{\epsilon}$ for any ${i}_{\epsilon}\in I$, the above equality (${d}_{u}({x}_{a},{x}_{b})=1$) shows that the net $\left({x}_{i}\right)$ does not converge to ${x}_{a}$. Consequently $\left({x}_{i}\right)$ 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

$({x}_{i}:i\in I)$ be a net in

X satisfying (

13). Let

${i}_{k}\u2a7e{i}_{k-1},\phantom{\rule{4pt}{0ex}}k\u2a7e2,$ be such that (

13) holds for

$\epsilon =1/{2}^{k},\phantom{\rule{4pt}{0ex}}k\in \mathbb{N}.$This is possible. Indeed, take

${i}_{1}$ such that (

13) holds for

$\epsilon =1/2.$ If

${i}_{2}^{\prime}$ is such that (

13) holds for

$\epsilon ={2}^{-2}$, then pick

${i}_{2}\in I$ such that

${i}_{2}\u2a7e{i}_{1},{i}_{2}^{\prime}.$ Continuing by induction one obtains the desired sequence

${\left({i}_{k}\right)}_{k\in \mathbb{N}}.$We distinguish two cases.

Case I.$\exists {j}_{0}\in I,\phantom{\rule{0.277778em}{0ex}}\exists {k}_{0}\in \mathbb{N},\phantom{\rule{1.em}{0ex}}\forall k\u2a7e{k}_{0},\phantom{\rule{0.277778em}{0ex}}{i}_{k}\u2a7d{j}_{0}\phantom{\rule{0.166667em}{0ex}}.$

Let $i\u2a7e{j}_{0}.$ Then for every k, ${i}_{k}\u2a7d{j}_{0}\u2a7di$ implies $d({x}_{i},{x}_{{j}_{0}})<{2}^{-k}$ so that $d({x}_{i},{x}_{{j}_{0}})=0.$ Since the quasi-metric space $(X,d)$ is ${T}_{1},$ it follows ${x}_{i}={x}_{{j}_{0}}$ for all $i\u2a7e{j}_{0}$ (see Proposition 1), so that the net $({x}_{i}:i\in I)$ is d-convergent to ${x}_{{j}_{0}}.$

Case II.$\forall j\in I,\phantom{\rule{0.277778em}{0ex}}\forall k\in \mathbb{N},\phantom{\rule{0.277778em}{0ex}}\exists {k}^{\prime}\u2a7ek,\phantom{\rule{1.em}{0ex}}{i}_{{k}^{\prime}}\nleqq j\phantom{\rule{0.166667em}{0ex}}.$

The inequalities $d({x}_{{i}_{k+1}},{x}_{{i}_{k}})<{2}^{-k},\phantom{\rule{4pt}{0ex}}k\in \mathbb{N},$ imply that the sequence ${\left({x}_{{i}_{k}}\right)}_{k\in \mathbb{N}}$ is right K-Cauchy (see Proposition 4), so it is d-convergent to some $x\in X.$

For $\epsilon >0$ choose ${k}_{0}\in \mathbb{N}$ such that ${2}^{-{k}_{0}}<\epsilon $ and $d(x,{x}_{{i}_{k}})<\epsilon $ for all $k\u2a7e{k}_{0}.$

Let

$i\in I,\phantom{\rule{0.166667em}{0ex}}i\u2a7e{i}_{{k}_{0}}.$ By hypothesis, there exists

$k\u2a7e{k}_{0}$ such that

${i}_{k}\nleqq i,$ implying

$i\u2a7d{i}_{k}\vee {i}_{k}\nsim i.$ Since

${i}_{{k}_{0}}\u2a7d{i}_{k},i$, by the choice of

${i}_{{k}_{0}}$,

$d({x}_{{i}_{k}},{x}_{i})<{2}^{-{k}_{0}}<\epsilon $ in both of these cases. But then

proving the convergence of the net

$\left({x}_{i}\right)$ 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.

**Proposition** **12.** A ${T}_{1}$ quasi-metric space $(X,d)$ is right K-sequentially complete if and only if every net satisfying the conditions (a) and (c) from Definition 6 is convergent.

**Proof.** Obviously, a proof is needed only for the case (c).

Suppose that the directed set

$(I,\u2a7d)$ has no maximal elements and let

$({x}_{i}:i\in I)$ 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 ${i}_{k}\u2a7d{i}_{k+1},\phantom{\rule{0.166667em}{0ex}}k\in \mathbb{N},$ be a sequence of indices in I such that $d({x}_{i},{x}_{j})<{2}^{-k}$ for all $i,j\u2a7e{i}_{k}$ with $i\nleqq j$. We show that we can further suppose that ${i}_{k+1}\nleqq {i}_{k}$.

Indeed, the fact that

I has no maximal elements implies that for every

$i\in I$ there exists

${i}^{\prime}\in I$ such that

Let

${i}_{1}^{\prime}\in I$ be such that (

10) holds for

$\epsilon ={2}^{-1}.$ Take

${i}_{1}$ such that

${i}_{1}^{\prime}\u2a7d{i}_{1}$ and

${i}_{1}\nleqq {i}_{1}^{\prime}$. Let

${i}_{2}^{\prime}\u2a7e{i}_{1}$ be such that (

10) holds for

$\epsilon ={2}^{-2}$ and let

${i}_{2}\in I$ satisfying

${i}_{2}^{\prime}\u2a7d{i}_{2}$ and

${i}_{2}\nleqq {i}_{2}^{\prime}$. Then

${i}_{1}\u2a7d{i}_{2}$ and

${i}_{2}\nleqq {i}_{1}$, because

${i}_{2}\u2a7d{i}_{1}\u2a7d{i}_{2}^{\prime}$ would contradict the choice of

${i}_{2}.$By induction one obtains a sequence

$\left({i}_{k}\right)$ in

I satisfying

${i}_{k}\u2a7d{i}_{k+1}$ and

${i}_{k+1}\nleqq {i}_{k}$ such that (

10) is satisfied with

$\epsilon ={2}^{-k}$ for every

${i}_{k}$.

We shall again consider two cases.

Case I.$\exists {j}_{0}\in I,\phantom{\rule{0.277778em}{0ex}}\exists {k}_{0}\in \mathbb{N},\phantom{\rule{1.em}{0ex}}\forall k\u2a7e{k}_{0},\phantom{\rule{0.277778em}{0ex}}{i}_{k}\u2a7d{j}_{0}\phantom{\rule{0.166667em}{0ex}}.$

Let

$i\u2a7e{j}_{0}$. By (

15) there exists

${i}^{\prime}\in I$ such that

$i\u2a7d{i}^{\prime}$ and

${i}^{\prime}\nleqq i,$ implying

$d({x}_{{i}^{\prime}},{x}_{i})<{2}^{-k}$ for all

$k\u2a7e{k}_{0}$, that is

$d({x}_{{i}^{\prime}},{x}_{i})=0$, so that, by

${T}_{1}$,

${x}_{{i}^{\prime}}={x}_{i}$.

We also have ${i}^{\prime}\nleqq {j}_{0}$ because ${i}^{\prime}\u2a7d{j}_{0}$ would imply ${i}^{\prime}\u2a7di$, in contradiction to the choice of ${i}^{\prime}$. But then, $d({x}_{{i}^{\prime}},{x}_{{j}_{0}})<{2}^{-k}$ for all $k\u2a7e{k}_{0}$, so that, as above, $d({x}_{{i}^{\prime}},{x}_{i})=0$ and ${x}_{{i}^{\prime}}={x}_{{j}_{0}}$.

Consequently, ${x}_{i}={x}_{{j}_{0}}$ for every $i\u2a7e{j}_{0}$, proving the convergence of the net $\left({x}_{i}\right)$ to ${x}_{{j}_{0}}$.

Case II.$\forall j\in I,\phantom{\rule{0.277778em}{0ex}}\forall k\in \mathbb{N},\phantom{\rule{0.277778em}{0ex}}\exists {k}^{\prime}\u2a7ek,\phantom{\rule{1.em}{0ex}}{i}_{{k}^{\prime}}\nleqq j\phantom{\rule{0.166667em}{0ex}}.$

The condition $d({x}_{{i}_{k}},{x}_{{i}_{k+1}})<{2}^{-k},\phantom{\rule{0.166667em}{0ex}}k\in \mathbb{N},$ implies that the sequence ${\left({x}_{{i}_{k}}\right)}_{k\in \mathbb{N}}$ is right K-Cauchy, so that there exists $x\in X$ with $d(x,{x}_{{i}_{k}})\to 0$ as $k\to \infty $.

For $\epsilon >0$ let ${k}_{0}\in \mathbb{N}$ be such that ${2}^{-{k}_{0}}<\epsilon $ and $d(x,{x}_{{i}_{k}})<\epsilon $ for all $k\u2a7e{k}_{0}$.

Let

$i\u2a7e{i}_{{k}_{0}}$. By II, for

$j=i$ and

$k={k}_{0}$, there exists

$k\u2a7e{k}_{0}$ such that

${i}_{k}\nleqq i$. The conditions

$k\u2a7e{k}_{0},\phantom{\rule{0.166667em}{0ex}}{i}_{{k}_{0}}\u2a7di,\phantom{\rule{0.166667em}{0ex}}{i}_{{k}_{0}}\u2a7d{i}_{k}$ and

${i}_{k}\nleqq i$ imply

so that

for all

$i\u2a7e{i}_{{k}_{0}}$. □