# Completeness in Quasi-Pseudometric Spaces—A Survey

## Abstract

**:**

## 1. Introduction

## 2. Metric and Uniform Spaces

**Definition**

**1.**

**Remark**

**1.**

**Theorem**

**1.**

- 1.
- The metric space X is complete.
- 2.
- Every Cauchy net in X is convergent.
- 3.
- Every Cauchy filter in $(X,{\mathcal{U}}_{d})$ is convergent.

**Theorem**

**2**

**.**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 ${F}_{n},\phantom{\rule{0.166667em}{0ex}}n\in \mathbb{N},$ of nonempty closed subsets of X

## 3. Quasi-Pseudometric and Quasi-Uniform Spaces

#### 3.1. Quasi-Pseudometric Spaces

**Example**

**1**

**.**For $x,y\in \mathbb{R}$ define a quasi-metric d by $d(x,y)=y-x,\phantom{\rule{0.166667em}{0ex}}$ if $x\le y$ and $d(x,y)=1$ if $x>y.$ A basis of open ${\tau}_{d}$-open neighborhoods of a point $x\in \mathbb{R}$ is formed by the family $[x,x+\epsilon ),\phantom{\rule{0.166667em}{0ex}}0<\epsilon <1.$ The family of intervals $(x-\epsilon ,x],\phantom{\rule{0.166667em}{0ex}}0<\epsilon <1,\phantom{\rule{0.166667em}{0ex}}$ forms a basis of open ${\tau}_{\overline{d}}\phantom{\rule{0.166667em}{0ex}}$-open neighborhoods of $x.$ Obviously, the topologies ${\tau}_{d}$ and ${\tau}_{\overline{d}}$ are Hausdorff and ${d}^{s}(x,y)=1$ for $x\ne y,$ so that $\tau \left({d}^{s}\right)$ is the discrete topology of $\mathbb{R}.$

**Asymmetric normed spaces**

**Example**

**2.**

**Proposition**

**1**

- 1.
- The ball ${B}_{d}(x,r)$ is d-open and the ball ${B}_{d}[x,r]$ is $\overline{d}$-closed. The ball ${B}_{d}[x,r]$ need not be d-closed.
- 2.
- The topology d is ${T}_{0}$ if and only if d is a quasi-metric.The topology d is ${T}_{1}$ if and only if $d(x,y)>0$ for all $x\ne y$ in X.
- 3.
- For every fixed $x\in X,$ the mapping $d(x,\xb7):X\to (\mathbb{R},|\xb7\left|\right)$ is d-upper semi-continuous and $\overline{d}$-lower semi-continuous.For every fixed $y\in X,$ the mapping $d(\xb7,y):X\to (\mathbb{R},|\xb7\left|\right)$ is d-lower semi-continuous and $\overline{d}$-upper semi-continuous.

- ${T}_{0}$ 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 $x\in X$ and closed set A not containing x, there exist the disjoint open sets $U,V$ such that $x\in U$ and $A\subseteq V.$

**Remark**

**2.**

#### 3.2. Quasi-Uniform Spaces

## 4. Cauchy Sequences and Sequential Completeness in Quasi-Pseudometric and Quasi-Uniform Spaces

**Definition**

**2.**

- left (right) d-Cauchy if for every $\epsilon >0$ there exist $x\in X$ and ${n}_{0}\in \mathbb{N}$ such that$$d(x,{x}_{n})<\epsilon \phantom{\rule{0.277778em}{0ex}}(respectively\phantom{\rule{0.277778em}{0ex}}d({x}_{n},x)<\epsilon )$$for all $n\u2a7e{n}_{0}$;
- ${d}^{s}$-Cauchy if it is a Cauchy sequence is the pseudometric space $(X,{d}^{s})$, that is for every $\epsilon >0$ there exists ${n}_{0}\in \mathbb{N}$ such that$${d}^{s}({x}_{n},{x}_{k})<\epsilon \phantom{\rule{0.277778em}{0ex}}for\text{}all\phantom{\rule{0.277778em}{0ex}}n,k\u2a7e{n}_{0}\phantom{\rule{0.166667em}{0ex}},$$a condition equivalent to$$d({x}_{n},{x}_{k})<\epsilon \phantom{\rule{0.277778em}{0ex}}for\text{}all\phantom{\rule{0.277778em}{0ex}}n,k\u2a7e{n}_{0}\phantom{\rule{0.166667em}{0ex}},$$as well as to$$\overline{d}({x}_{n},{x}_{k})<\epsilon \phantom{\rule{0.277778em}{0ex}}for\text{}all\phantom{\rule{0.277778em}{0ex}}n,k\u2a7e{n}_{0}\phantom{\rule{0.166667em}{0ex}};$$
- left (right) K-Cauchy if for every $\epsilon >0$ there exists ${n}_{0}\in \mathbb{N}$ such that$$d({x}_{k},{x}_{n})<\epsilon \phantom{\rule{0.277778em}{0ex}}(respectively\phantom{\rule{0.277778em}{0ex}}d({x}_{n},{x}_{k})<\epsilon )$$for all $n,k\in \mathbb{N}$ with ${n}_{0}\u2a7dk\u2a7dn$;
- weakly left (right) K-Cauchy if for every $\epsilon >0$ there exists ${n}_{0}\in \mathbb{N}$ such that$$d({x}_{{n}_{0}},{x}_{n})<\epsilon \phantom{\rule{0.277778em}{0ex}}(respectively\phantom{\rule{0.277778em}{0ex}}d({x}_{n},{x}_{{n}_{0}})<\epsilon )\phantom{\rule{0.166667em}{0ex}},$$for all $n\u2a7e{n}_{0}\phantom{\rule{0.166667em}{0ex}}.$

**Remark**

**4**

- 1.
- These notions are related in the following way:${d}^{s}$-Cauchy ⇒ left K-Cauchy $\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}$ weakly left K-Cauchy $\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}$ left d-Cauchy.The same implications hold for the corresponding right notions. No one of the above implications is reversible.
- 2.
- 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 $\overline{d}.$
- 3.
- A sequence is ${d}^{s}$-Cauchy if and only if it is both left and right d-K-Cauchy.
- 4.
- A d-convergent sequence is left d-Cauchy and a $\overline{d}$-convergent sequence is right d-Cauchy. For the other notions, a convergent sequence need not be Cauchy.
- 5.
- If each convergent sequence in a regular quasi-metric space $(X,d)$ admits a left K-Cauchy subsequence, then X is metrizable ([18]).

**Proposition**

**2**

**.**Let $\left({x}_{n}\right)$ be a left or right K-Cauchy sequence in a quasi-pseudometric space $(X,d).$

- 1.
- If $\left({x}_{n}\right)$ has a subsequence which is d-convergent to $x,$ then $\left({x}_{n}\right)$ is d-convergent to $x.$
- 2.
- If $\left({x}_{n}\right)$ has a subsequence which is $\overline{d}$-convergent to $x,$ then $\left({x}_{n}\right)$ is $\overline{d}$-convergent to $x.$
- 3.
- If $\left({x}_{n}\right)$ has a subsequence which is ${d}^{s}$-convergent to x, then $\left({x}_{n}\right)$ is ${d}^{s}$-convergent to x.

**Definition**

**3**

- sequentially d-complete if every ${d}^{s}$-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 ${d}^{s}$-convergent;
- 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.

**Remark**

**5.**

**Proposition**

**3**

**.**A quasi-pseudometric space is sequentially weakly left K-complete if and only if it is sequentially left K-complete.

**Proposition**

**4**

- 1.
- If a sequence $\left({x}_{n}\right)$ in X satisfies ${\sum}_{n=1}^{\infty}d({x}_{n},{x}_{n+1})$$<\infty $ $({\sum}_{n=1}^{\infty}d({x}_{n+1},{x}_{n})<\infty ),$ then it is left (right) d-K-Cauchy.
- 2.
- The quasi-pseudometric space $(X,d)$ is sequentially left (right) d-K-complete if and only if every sequence $\left({x}_{n}\right)$ in X satisfying ${\sum}_{n=1}^{\infty}d({x}_{n},{x}_{n+1})<\infty $ (resp. ${\sum}_{n=1}^{\infty}d({x}_{n+1},{x}_{n})<\infty )$ is d-convergent.
- 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**

**.**

**Theorem**

**3**

**.**A quasi-pseudometric space $(X,d)$ is sequentially d-complete if and only if each decreasing sequence ${F}_{1}\supseteq {F}_{2}\dots $ of nonempty closed sets with $diam\left({F}_{n}\right)\to 0$ as $n\to \infty $ has nonempty intersection, which is a singleton if d is a quasi-metric.

**Proposition**

**5.**

## 5. Completeness by Nets and Filters

#### 5.1. Some Positive Results

**Proposition**

**6**

- 1.
- Every left d-K-Cauchy sequence is ${d}^{s}$-convergent.
- 2.
- Every left d-K-Cauchy net is ${d}^{s}$-convergent.

**Definition**

**4.**

- 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 $x\in F$.

- 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$$\phantom{\rule{1.em}{0ex}}\forall U\in \mathcal{U},\phantom{\rule{0.277778em}{0ex}}\exists {i}_{0}\in I,\phantom{\rule{0.277778em}{0ex}}\forall i,j\in I,\phantom{\rule{0.277778em}{0ex}}{i}_{0}\u2a7di\u2a7dj\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}({x}_{i},{x}_{j})\in U\phantom{\rule{1.em}{0ex}}(resp.\phantom{\rule{0.277778em}{0ex}}({x}_{j},{x}_{i})\in U\phantom{\rule{0.166667em}{0ex}}.$$

**Definition**

**5.**

- 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.

**Remark**

**6.**

**Proposition**

**7**

- 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.

**Proposition**

**8**

**.**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

**Proposition**

**9**

- 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**

**Proposition**

**10.**

**Proof.**

**Stoltenberg-Cauchy nets**

**Gregori-Ferrer-Cauchy nets**

**Example**

**3**

**.**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 $\mathbb{N}$ with

**Definition**

**6.**

- (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**

- strictly maximal if there is no $i\in I\backslash \left\{j\right\}$ with $j\u2a7di,$ or, equivalently,$$j\u2a7di\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}i=j,\phantom{\rule{1.em}{0ex}}\mathrm{for}\text{}\mathrm{every}\phantom{\rule{0.277778em}{0ex}}i\in I;$$
- maximal if$$j\u2a7di\phantom{\rule{0.277778em}{0ex}}\Rightarrow \phantom{\rule{0.277778em}{0ex}}i\u2a7dj,\phantom{\rule{1.em}{0ex}}\mathrm{for}\text{}\mathrm{every}\phantom{\rule{0.277778em}{0ex}}i\in I\phantom{\rule{0.166667em}{0ex}}.$$

**Remark**

**7.**

- 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.**

**Remark**

**8.**

- 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.**

- 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 $d(x,{x}_{i})=0<\epsilon .$

**Strongly Stoltenberg-Cauchy nets**

**Definition**

**7.**

**Remark**

**9.**

**Proof.**

**Example**

**4.**

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

**Proposition**

**11**

**Proof.**

**The proof of Proposition 11 in the case of GF-completeness**

**Proposition**

**12.**

**Proof.**

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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Cobzas, Ş.
Completeness in Quasi-Pseudometric Spaces—A Survey. *Mathematics* **2020**, *8*, 1279.
https://doi.org/10.3390/math8081279

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Cobzas Ş.
Completeness in Quasi-Pseudometric Spaces—A Survey. *Mathematics*. 2020; 8(8):1279.
https://doi.org/10.3390/math8081279

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Cobzas, Ştefan.
2020. "Completeness in Quasi-Pseudometric Spaces—A Survey" *Mathematics* 8, no. 8: 1279.
https://doi.org/10.3390/math8081279