2. Quasi-Metric Spaces

Let (X,d) be a quasi-metric space, $(x_{i})_{i\in I,\sqsubseteq}$ be a net, and $x\in X$ . If for all $y\in X$ , it holds that $d(x,y)=\limsup d(x_{i},y)$ , then $x\in X$ is called a d-limit of $(x_{i})_{i\in I,\sqsubseteq}$ , denoted by $x=\lim_{d} x_{i}$ . A net $(x_{i},r_{i})_{i\in I,\sqsubseteq}$ in $X\times [0,+\infty)$ is Cauchy-weighted in (X,d) if $\bigwedge_{i\in I}r_{i}=0$ , and for each $i, i^{'}\in I$ , whenever $i\sqsubseteq i^{'}$ , $d(x_{i}, x_{i^{'}})\leq r_{i}-r_{i^{'}}$ . For any Cauchy-weighted net $(x_{i},r_{i})_{i\in I,\sqsubseteq}$ in (X,d), $x\in X$ is a d-limit of the Cauchy net $(x_{i})_{i\in I,\sqsubseteq}$ if and only if for any $y\in X$ , it holds that $d(x,y)=\bigvee_{i\in I}(d(x_{i},y)-r_{i})$ .

Definition 2 (Goubault-Larrecq Reference Goubault-Larrecq2013). Let (X,d) be a quasi-metric space. Define ${\bf B}(X,d)=X\times[0,+\infty)$ . We call the elements of ${\bf B}(X,d)$ formal balls. On ${\bf B}(X,d)$ , we can define a quasi-metric $d^{+}$ as follows:

$$\forall\ (x,r),(y,s)\in {\bf B}(X,d),\ d^{+}((x,r),(y,s))=\max(d(x,y)-r+s,0),$$

and as a result the induced order $\leq^{d^{+}}$ defined by

$$\forall\ (x,r),(y,s)\in {\bf B}(X,d),\ (x,r)\leq^{d^{+}}(y,s)\Longleftrightarrow d(x,y)\leq r-s.$$

Unless otherwise stated, throughout the paper, whenever an order is mentioned in the context of ${\bf B}(X,d)$ , it is to be interpreted with respect to the induced order $\leq^{d^{+}}$ on ${\bf B}(X,d)$ .

A quasi-metric space (X,d) is standard if and only if, for every directed family of formal balls $(x_{i},r_{i})_{i\in I}$ , for every $s\in [0,+\infty)$ , $(x_{i},r_{i})_{i\in I}$ has a supremum in ${\bf B}(X,d)$ if and only if $(x_{i},r_{i}+s)_{i\in I}$ has a supremum in ${\bf B}(X,d)$ .

Let (X,d) be a quasi-metric space. A directed family $(x_{i},r_{i})_{i\in I}$ in ${\bf B}(X,d)$ is translational complete if for any $a\in[\!-\bigwedge_{i\in I}r_{i},+\infty)$ , $\bigvee_{i\in I}(x_{i},r_{i}+a)$ exists.

Let (X,d) and $(X^{'}\!,d^{'})$ be two quasi-metric spaces. A map $g: {\bf B}(X,d)\longrightarrow {\bf B}(X^{'}\!,d^{'})$ is Y-Scott continuous if it preserves the suprema of translational complete directed families.

Proposition 4 (Ng and Ho Reference Ng and Ho2017). Let (X,d) and $(X^{'}\!,d^{'})$ be two quasi-metric spaces, and $f: X\longrightarrow X^{'}$ be a mapping. Define a mapping ${\bf B}(f): {\bf B}(X,d)\longrightarrow {\bf B}(X^{'}\!,d^{'})$ as follows:

$$\forall\ (x,r)\in {\bf B}(X,d),\ {\bf B}(f)(x,r)=(f(x),r).$$

Then f is Y-continuous if and only if ${\bf B}(f)$ is a Y-Scott continuous map.

We denote the collection of all g-closed subsets of ${\bf B}(X,d)$ by $\Gamma_{g}({\bf B}(X,d))$ . Then $\Gamma_{g}({\bf B}(X,d))$ is a co-topology on ${\bf B}(X,d)$ , that is, the family

$$\{U\subseteq {\bf B}(X,d)\mid \mbox{the complement of } U\ \mbox{is a } g\mbox{-closed subset of}\ {\bf B}(X,d)\}$$

is a topology, called the g-topology on ${\bf B}(X,d)$ . For any subset E of B(X,d), let $cl_{g}(E)$ denote the g-closure of E. Define the Hausdorf-f-Hoare quasi-metric $d_{\mathcal{H}}$ on $\Gamma_{g}({\bf B}(X,d))$ by $d_{\mathcal{H}}(A,B)=\bigvee_{(a,m)\in A}\bigwedge_{(c,r)\in B}d^{+}((a,m),(c,r)).$ Then $d_{\mathcal{H}}(A,B)=0$ if and only if $A\subseteq B$ .

3. Local Yoneda-Complete Quasi-Metric Spaces

Example 11. (1) $([0,+\infty),d)$ is local Yoneda-complete, not Yoneda-complete, where $d: [0,+\infty)\times [0,+\infty)\longrightarrow[0,+\infty]$ as follows:

\begin{eqnarray*} d(x,y)=\left\{ \begin{array}{l@{\quad}l} 0,&{x\le y,}\\ {x - y,} & {y < x.}\end{array}\right. \end{eqnarray*}

In fact, the map $(x,r)\longmapsto (x\!-\!r,-r)$ defines an order isomorphism from ${\bf B}([0,+\infty),d)$ onto $C=\{(a,b)\in\mathbb{R}\times(\!-\infty,0]\mid a\geq b\}$ ordered component-wise. Since C is a local dcpo, we have that ${\bf B}([0,+\infty),d)$ is a local dcpo. Let $(x_{i},r_{i})_{i\in I}$ be a directed family in ${\bf B}([0,+\infty),d)$ with supremum (x,r). Then $\bigvee\limits_{i\in I}^{C}(x_{i}\!-\!r_{i},-r_{i})=(x-r,-r)$ , and thus $\bigvee_{i\in I}(\!-r_{i})=-r$ . Hence, $r=\bigwedge_{i\in I}r_{i}$ . So we conclude that $([0,+\infty),d)$ is local Yoneda-complete. Obviously, C is not a dcpo. Then ${\bf B}([0,+\infty),d)$ is not a dcpo, and thus $([0,+\infty),d)$ is not Yoneda-complete.

(2) Every poset $(X,\leq)$ can be seen as a quasi-metric space by letting

\begin{eqnarray*} d_{\leq}(x,y)=\left\{ \begin{array}{l@{\quad}l} 0,&{x\leq y,}\\ +\infty,&{\rm otherwise}.\end{array}\right. \end{eqnarray*}

On formal balls $(x,r)\leq^{d_{\leq}\!{^{+}}}(y,s)$ if and only if $x\leq y$ and $r\geq s$ . So $(x,r)\longmapsto (x,-r)$ defines an order isomorphism from ${\bf B}(X,d_{\leq})$ onto $X\times(\!-\infty,0]$ ordered component-wise. Obviously, $(X,d_{\leq})$ satisfies the condition (1) in Definition 9. This implies that $(X,d_{\leq})$ is a local Yoneda-complete quasi-metric space if and only if X is a local dcpo. Since $\mathbb{N}$ is a local dcpo, not a dcpo, we have that $(\mathbb{N}, d_{\preceq})$ is local Yoneda-complete, not Yoneda-complete, where $\mathbb{N}$ denotes the set of natural numbers with the usual order $\preceq$ . Moreover, let $X=\mathbb{N}\cup\{\omega_{1},\omega_{2}\}$ , and let the partial order $\leq$ on X as follows:

$\bullet$ for $x\in\mathbb{N}$ , $x\leq \omega_{1},\omega_{2}$ ;

$\bullet$ $\omega_{1}\leq\omega_{1}$ , $\omega_{2}\leq\omega_{2}$ ;

$\bullet$ for $x,y\in \mathbb{N}$ , $x\leq y$ if and only if $x\preceq y$ .

That is, we add $\omega_{1},\omega_{2}$ above all elements in $\mathbb{N}$ . Then X is not a local dcpo, and thus $(X,d_{\leq})$ does not satisfy the condition (2) in Definition 9.

(3) Let $X=\{0\}\cup\{\frac{1}{2^{m}}\mid m\in\mathbb{N}_{+}\}$ and d on $X\times X$ which is defined as follows:

\begin{eqnarray*} d(x,y)=\left\{ \begin{array}{l@{\quad}l} 0,&{x=y,}\\+\infty,&{x<y,}\\\sqrt{2},&{0=y<x,}\\ x-y,&{0<y<x,}\end{array}\right. \end{eqnarray*}

where $\mathbb{N}_{+}$ denotes the set of all positive integers. Then (X,d) is a quasi-metric space. Let $(x,r), (y,s)\in {\bf B}(X,d)$ . If $(x,r)\leq^{d{^{+}}}(y,s)$ , then $d(x,y)\leq r-s$ , and thus $x\geq y$ . First, we shall prove that ${\bf B}(X,d)$ is a local dcpo. Let $(x_{i},r_{i})_{i\in I}$ be a directed family in ${\bf B}(X,d)$ with an upper bound (x,r). Then $d(x_{i},x)\leq r_{i}\!-\!r$ for any $i\in I$ . There are three cases:

Consequently, ${\bf B}(X,d)$ is a local dcpo. Yet since $\bigvee_{m\in\mathbb{N}}(\frac{1}{2^{m}},\frac{1}{2^{m}}+\sqrt{2})=(0,0)$ as shown in Case 2 and $\bigwedge_{m\in\mathbb{N}}(\frac{1}{2^{m}}+\sqrt{2})=\sqrt{2}$ , we have that (X,d) does not satisfy the condition (1) in Definition 9.

Necessity. Let (X,d) be a local Yoneda-complete metric space. We shall prove that ${\bf B}(X,d)$ is a dcpo. Let $(x_{i},r_{i})_{i\in I}$ be a directed family in ${\bf B}(X,d)$ and let $r_{\infty}=\bigwedge_{i\in I}r_{i}$ . Then there exists $(y_{n},s_{n})\in(x_{i},r_{i})_{i\in I}$ such that $s_{n}<r_{\infty}+\frac{1}{n}$ for any $n\in \mathbb{N}_{+}$ , where $\mathbb{N}_{+}$ denotes the set of all positive integers. Let $(x_{1},r_{1})=(y_{1},s_{1})$ . For any $n>1$ , let $(x_{n},r_{n})$ be an upper bound of $(y_{n},s_{n})$ and $(x_{n-1},r_{n-1})$ . Then $(x_{1},r_{1})\leq^{d{^{+}}}(x_{2},r_{2})\leq^{d{^{+}}}\cdot\cdot\cdot\leq^{d{^{+}}}(x_{n},r_{n})\leq^{d{^{+}}}\cdot\cdot\cdot,$ and thus $\cdot\cdot\cdot \leq r_{n}\leq r_{n-1}\leq\cdot\cdot\cdot\leq r_{2}\leq r_{1}$ . For a fixed $\varepsilon>0$ , there exists $n_{0}\in \mathbb{N}_{+}$ such that $r_{n}-r_{m}<\varepsilon$ for any $n_{0}\leq n\leq m$ . For all n in $\mathbb{N}_{+}$ , there exists $l\in \mathbb{N}_{+}$ such that $n_{0},n\leq l$ , and by symmetry $d(x_{l},x_{n_{0}})\leq r_{n_{0}}-r_{l}<\varepsilon$ , and thus $d(x_{n},x_{n_{0}})\leq d(x_{n},x_{l})+d(x_{l},x_{n_{0}})< r_{n}-r_{l}+\varepsilon\leq r_{n}+\varepsilon.$ Hence $(x_{n},r_{n}+\varepsilon)\leq^{d{^{+}}} (x_{n_{0}},0)$ . So we conclude that $(x_{n_{0}},0)$ is an upper bound of the directed family $(x_{n},r_{n}+\varepsilon)_{n\in\mathbb{N}_{+}}$ . By hypothesis, we have that $\bigvee_{n\in\mathbb{N}_{+}}(x_{n},r_{n}+\varepsilon)$ exists, denoted by (x,r). Then $r=\bigwedge_{n\in\mathbb{N}_{+}}(r_{n}+\varepsilon)$ . Obviously, $(x,r-\varepsilon)$ is an upper bound of $(x_{n},r_{n})_{n\in\mathbb{N}_{+}}$ . Let (y,s) be an upper bound of $(x_{n},r_{n})_{n\in\mathbb{N}_{+}}$ . Then $(x_{n},r_{n}+\varepsilon)\leq^{d{^{+}}}(y,s+\varepsilon)$ for any $n\in\mathbb{N}_{+}$ , and thus $(x,r)\leq^{d{^{+}}}(y,s+\varepsilon)$ . Hence $(x,r-\varepsilon)\leq^{d{^{+}}}(y,s)$ , and so we conclude that $\bigvee_{n\in\mathbb{N}_{+}}(x_{n},r_{n})=(x,r-\varepsilon)$ . Let $i\in I$ . Then there exists $(z_{n},t_{n})\in (x_{i},r_{i})_{i\in I}$ such that $(x_{n},r_{n}),(x_{i},r_{i})\leq^{d{^{+}}}(z_{n},t_{n})$ for any $n\in \mathbb{N}_{+}$ . Hence

\begin{align*}\begin{array}{lll}d(x_{i},x)\leq d(x_{i},z_{n})+d(z_{n},x_{n})+d(x_{n},x)&\leq& r_{i}-t_{n}+r_{n}-t_{n}+r_{n}-(r-\varepsilon)\\[3pt]&=&r_{i}-(r-\varepsilon)+2(r_{n}-t_{n})\\[3pt]&\leq&r_{i}-(r-\varepsilon)+2(r_{n}-r_{\infty})\\[3pt]&<&r_{i}-(r-\varepsilon)+\frac{2}{n}.\end{array}\end{align*}

This implies $d(x_{i},x)\leq r_{i}-(r-\varepsilon)$ . Then $(x_{i},r_{i})\leq^{d{^{+}}}(x,r-\varepsilon)$ , and thus $(x,r-\varepsilon)$ is an upper bound of $(x_{i},r_{i})_{i\in I}$ . Suppose that (z,t) is an upper bound of $(x_{i},r_{i})_{i\in I}$ . Then (z,t) is an upper bound of $(x_{n},r_{n})_{n\in\mathbb{N}_{+}}$ , and thus $(x,r-\varepsilon)\leq^{d{^{+}}}(z,t)$ . Hence, $\bigvee_{i\in I}(x_{i},r_{i})=(x,r-\varepsilon)$ . This shows that ${\bf B}(X,d)$ is a dcpo. Therefore, (X,d) is complete.

Proof. Necessity. Let (X,d) be a quasi-metric space where every local net has a d-limit. We start by proving part (2) of Definition 9. Let $(x_{i},r_{i})_{i\in I}$ be a directed family in ${\bf B}(X,d)$ with an upper bound. Define a preorder $\sqsubseteq$ on I as follows:

$$\forall\ i,j\in I, i\sqsubseteq j \Longleftrightarrow (x_{i},r_{i})\leq^{d{^{+}}}(x_{j},r_{j}).$$

Then $(x_{i},r_{i})_{i\in I,\sqsubseteq}$ is a monotone net in ${\bf B}(X,d)$ and has an upper bound, and thus $(x_{i})_{i\in I,\sqsubseteq}$ is a local net. Hence, the net $(x_{i})_{i\in I,\sqsubseteq}$ has a d-limit y. This implies $\bigwedge_{i\in I}\bigvee_{j\in I,i\sqsubseteq j}d(x_{j},y)=0$ . Then for all $\varepsilon>0$ , there exists $i_{0}\in I$ such that $\bigvee_{j\in I,i_{0}\sqsubseteq j}d(x_{j},y)<\varepsilon$ . Let $i\in I$ . Then there exists $j\in I$ such that $i, i_{0}\sqsubseteq j$ , and thus

$$\begin{array}{lll}d(x_{i},y)\leq d(x_{i},x_{j})+d(x_{j},y)&\leq& r_{i}-r_{j}+d(x_{j},y)\\[3pt]&\leq& r_{i}-\bigwedge_{i\in I}r_{i}+d(x_{j},y)\\[3pt]&<&r_{i}-\bigwedge_{i\in I}r_{i}+\varepsilon.\end{array}$$

So we conclude that $d(x_{i},y)\leq r_{i}-\bigwedge_{i\in I}r_{i}$ . Hence, $(y,\bigwedge_{i\in I}r_{i})$ is an upper bound of $(x_{i},r_{i})_{i\in I}$ . Let (z,s) be an upper bound of $(x_{i},r_{i})_{i\in I}$ . Then $d(x_{i},z)\leq r_{i}-s$ for any $i\in I$ , and thus $s\leq\bigwedge_{i\in I}r_{i}$ . Since $d(y,z)=\bigwedge_{i\in I}\bigvee_{j\in I,i\sqsubseteq j}d(x_{j},z)\leq\bigwedge_{i\in I}\bigvee_{j\in I,i\sqsubseteq j}(r_{j}-s)\leq\bigwedge_{i\in I}(r_{i}-s)=(\bigwedge_{i\in I}r_{i})-s$ , we have that $(y,\bigwedge_{i\in I}r_{i})\leq^{d{^{+}}}(z,s)$ . This shows that $\bigvee_{i\in I}(x_{i},r_{i})=(y,\bigwedge_{i\in I}r_{i})$ . Part (1) of Definition 9 is then a trivial consequence. Therefore, (X,d) is a local Yoneda-complete quasi-metric space.

Sufficiency. Let (X,d) be a local Yoneda-complete quasi-metric space, and let $(x_{i})_{i\in I,\sqsubseteq}$ be a local net. Then, there exists $(r_{i})_{i\in I,\sqsubseteq}\subseteq[0,+\infty)$ such that $(x_{i},r_{i})_{i\in I,\sqsubseteq}$ is a monotone net in ${\bf B}(X,d)$ and has an upper bound, and thus $(x_{i},r_{i})_{i\in I}$ is a directed family in ${\bf B}(X,d)$ and has an upper bound. Hence $\bigvee_{i\in I}(x_{i},r_{i})$ exists, denoted by (x,r). By hypothesis, we have that $r=\bigwedge_{i\in I}r_{i}$ . Then $(x_{i},r_{i}-\bigwedge_{i\in I}r_{i})_{i\in I,\sqsubseteq}$ is a Cauchy-weighted net and $\bigvee_{i\in I}(x_{i},r_{i}-\bigwedge_{i\in I}r_{i})=(x,0)$ . Next, we shall prove that x is the d-limit of $(x_{i})_{i\in I,\sqsubseteq}$ . For any $z\in X$ , since $d(x_{i},z)\leq d(x_{i},x)+d(x,z)\leq r_{i}-\bigwedge_{i\in I}r_{i}+d(x,z)$ for any $i\in I$ , we have that $d(x_{i},z)-r_{i}\leq d(x,z)-\bigwedge_{i\in I}r_{i}$ for any $i\in I$ . Obviously, $\bigvee_{i\in I}(d(x_{i},z)-r_{i})$ exists, denoted by s. Then $s\leq d(x,z)-\bigwedge_{i\in I}r_{i}$ . Suppose that $s<d(x,z)-\bigwedge_{i\in I}r_{i}$ . Then $s<+\infty$ . Since $d(x_{i},z)-r_{i}\leq s$ for any $i\in I$ , we have that $d(x_{i},z)\leq r_{i}+s$ for any $i\in I$ . Then $(x_{i},r_{i}+s)\leq^{d{^{+}}}(z,0)$ for any $i\in I$ , and thus $\bigvee_{i\in I}(x_{i},r_{i}+s)$ exists, denoted by $(y,\bigwedge_{i\in I}r_{i}+s)$ . Hence $y=x$ , that is $\bigvee_{i\in I}(x_{i},r_{i}+s)=(x,\bigwedge_{i\in I}r_{i}+s)$ . This implies $(x,\bigwedge_{i\in I}r_{i}+s)\leq^{d{^{+}}}(z,0)$ . Then $d(x,z)\leq\bigwedge_{i\in I}r_{i}+s$ , which is a contradiction. So we conclude that $s=d(x,z)-\bigwedge_{i\in I}r_{i}$ . Therefore, $d(x,z)=\bigvee_{i\in I}(d(x_{i},z)-r_{i})+\bigwedge_{i\in I}r_{i}=\bigvee_{i\in I}(d(x_{i},z)-r_{i}+\bigwedge_{i\in I}r_{i})$ . This shows that x is the d-limit of $(x_{i})_{i\in I,\sqsubseteq}$ .

4. The Local Yoneda Completions of Quasi-Metric Spaces

Definition 16. A local Yoneda completion of a quasi-metric space (X,d) is a local Yoneda-complete quasi-metric space $(\hat{X},\hat{d})$ , together with a Y-continuous map $\tau: X\longrightarrow\hat{X}$ , such that for any local Yoneda-complete quasi-metric space $(X^{'}\!,e)$ and Y-continuous map $f: X\longrightarrow X^{'}$ , there exists a unique Y-continuous map $\hat{f}:\hat{X}\longrightarrow X^{'}$ such that $f=\hat{f}\circ\tau$ , i.e., the following diagram commutes:

Let $\tilde{X}=\{A\subseteq {\bf B}(X,d)\mid A\mbox{ is a $g$-closed set satisfying }\alpha(A)=0\}$ , where the notion of g-closed sets is introduced in Definition 5. Define a mapping $\tilde{d}=d_{\mathcal{H}}\mid_{\tilde{X}\times\tilde{X}}:\tilde{X}\times\tilde{X}\longrightarrow[0,+\infty]$ as follows:

$$\forall\ (A,B)\in\tilde{X}\times\tilde{X},\ \tilde{d}(A,B)=\bigvee_{(a,m)\in A}\bigwedge_{(c,r)\in B}d^{+}((a,m),(c,r)).$$

Then $(\tilde{X},\tilde{d})$ is a Yoneda-complete quasi-metric space (see Ng and Ho Reference Ng and Ho2017). We write

$$\hat{X}=\{A\subseteq {\bf B}(X,d)\mid A \mbox{ is a $g$-closed local $g$-set satisfying }\alpha(A)=0\}.$$

Then $\hat{X}\subseteq\tilde{X}.$ Define the mapping $\hat{d}$ as the restriction of $\tilde{d}$ to $\hat{X}\times\hat{X}$ . Then $(\hat{X},\hat{d})$ is a quasi-metric space.

Proof. Obviously, $cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i}))$ is a g-closed set. Since

$$\begin{array}{lll}\alpha(cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i})))&\leq&\alpha(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i}))\\&=&\bigwedge_{i\in I}\alpha(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i})\\&=&\bigwedge_{i\in I}(r_{i}-\bigwedge_{i\in I}r_{i})\\&=&0,\end{array}$$

we have that $cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i}))\in\tilde{X}$ . Since $(A_{i},r_{i})\leq^{\tilde{d}{^{+}}}(cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i})),\bigwedge_{i\in I}r_{i})$ for any $i\in I$ , we have that $(cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i})),\bigwedge_{i\in I}r_{i})$ is an upper bound of $(A_{i},r_{i})_{i\in I}$ in ${\bf B}(\tilde{X},\tilde{d})$ . Let (B,s) be an upper bound of $(A_{i},r_{i})_{i\in I}$ in ${\bf B}(\tilde{X},\tilde{d})$ . Then $(A_{i},r_{i})\leq^{\tilde{d}{^{+}}} (B,s)$ for any $i\in I$ , and thus $A_{i}+r_{i}\subseteq B+s$ for any $i\in I$ . So we conclude that $\bigcup_{i\in I}(A_{i}+r_{i})\subseteq B+s$ , and hence $cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}))\subseteq B+s.$ Since $\alpha(\bigcup_{i\in I}(A_{i}+r_{i}))=\bigwedge_{i\in I}r_{i}$ , it follows from Proposition 7 that $cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}))=cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i}))+\bigwedge_{i\in I}r_{i}.$ Therefore, $cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i}))+\bigwedge_{i\in I}r_{i}\subseteq B+s$ , whence $(cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i})),\bigwedge_{i\in I}r_{i})\leq^{\tilde{d}{^{+}}}(B,s)$ by Proposition 19. Therefore, $\bigvee\limits_{i\in I}^{{\bf B}(\tilde{X},\tilde{d})}(A_{i},r_{i})=(cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i})),\bigwedge_{i\in I}r_{i})$ .

Next, we show that the sub-poset ${\bf B}(\hat{X},\hat{d})$ of ${\bf B}(\tilde{X},\tilde{d})$ is closed under least upper bounds of bounded directed families.

Let $\mathcal{A}\subseteq\tilde{X}$ . $\mathcal{A}$ satisfies Condition $(\!\ast\!)$ if, whenever for any directed family $(A_{i},r_{i})_{i\in I}$ in ${\bf B}(\mathcal{A},d_{\mathcal{H}}\mid_{\mathcal{A}\times \mathcal{A}})$ with an upper bound (Z,t), then

$$\bigvee\limits_{i\in I}^{{\bf B}(\tilde{X},\tilde{d})}(A_{i},r_{i})\in {\bf B}(\mathcal{A},d_{\mathcal{H}}\mid_{\mathcal{A}\times \mathcal{A}}\!).$$

Proposition 21 means that $\hat{X}$ satisfies Condition $(\!\ast\!)$ . Let $\Psi(X)=\{\mathord{\downarrow}\hspace{0.05em}(x,0)\mid x\in X\}$ , and

$$cl_{L}(\Psi(X))=\bigcap\{\mathcal{A}\subseteq\tilde{X}\mid \Psi(X)\subseteq\mathcal{A} \mbox{ and } \mathcal{A} \mbox{ satisfies } {\bf Condition} (\!\ast\!)\}.$$

Define the mapping $d_{L}$ as the restriction of $\tilde{d}$ to $cl_{L}(\Psi(X))$ . Then $(cl_{L}(\Psi(X)),d_{L})$ is a quasi-metric space.

Proposition 23. Let (X,d) be a quasi-metric space. Define a mapping $\zeta: X\longrightarrow cl_{L}(\Psi(X))$ as follows:

$$\forall\ x\in X,\ \zeta(x)=\mathord{\downarrow}\hspace{0.05em}(x,0).$$

Then $\zeta$ is a Y-continuous mapping.

Proof. Let $(x,r),(y,s)\in {\bf B}(X,d)$ satisfying $(x,r)\leq^{d{^{+}}}(y,s)$ . Then $d(x,y)\leq r-s$ . If $(a,m)\in\mathord{\downarrow}\hspace{0.05em}(x,0)$ , then $d^{+}((a,m),(x,0))=0$ , and thus

$$\bigwedge\limits_{(b,n)\in\mathord{\downarrow}\hspace{0.05em}(y,0)}d^{+}((a,m),(b,n))\leq d^{+}((a,m),(y,0))\leq d^{+}((x,0),(y,0))\leq r-s.$$

Hence $\bigvee\limits_{(a,m)\in\mathord{\downarrow}\hspace{0.05em}(x,0)}\bigwedge\limits_{(b,n)\in\mathord{\downarrow}\hspace{0.05em}(y,0)}d^{+}((a,m),(b,n))\leq r-s$ . So we conclude that

$$d_{L}(\mathord{\downarrow}\hspace{0.05em}(x,0),\mathord{\downarrow}\hspace{0.05em}(y,0))\leq r-s.$$

Therefore, $(\mathord{\downarrow}\hspace{0.05em}(x,0),r)\leq^{{d_{L}}\!{^{+}}}(\mathord{\downarrow}\hspace{0.05em}(y,0),s)$ . This shows that ${\bf B}(\zeta)(x,r)\leq^{{d_{L}}\!{^{+}}} {\bf B}(\zeta)(y,s).$

Let $(x_{i},r_{i})_{i\in I}$ be a translational complete directed family in ${\bf B}(X,d)$ with supremum (x,r). Then $r=\bigwedge_{i\in I}r_{i}$ , and $(\mathord{\downarrow}\hspace{0.05em}(x_{i},0),r_{i})_{i\in I}$ is a directed family in ${\bf B}(cl_{L}(\Psi(X)),d_{L})$ . Since $(x_{i},r_{i})\leq^{d{^{+}}} (x,r)$ for any $i\in I$ , we have that $(\mathord{\downarrow}\hspace{0.05em}(x_{i},0),r_{i})\leq^{{d_{L}}\!{^{+}}}(\mathord{\downarrow}\hspace{0.05em}(x,0),r)$ for any $i\in I$ . Let $(B,s)\in {\bf B}(cl_{L}(\Psi(X)),d_{L})$ such that $(\mathord{\downarrow}\hspace{0.05em}(x_{i},0),r_{i})\leq^{{d_{L}}\!{^{+}}} (B,s)$ for any $i\in I$ . Then $s\leq\bigwedge_{i\in I}r_{i}$ , and $(x_{i},r_{i})\in B+s$ for any $i\in I$ . Since $B+s$ is a g-closed set, we have that $(x,r)\in B+s$ . Then $(x,r-s)\in B$ , and thus $\mathord{\downarrow}\hspace{0.05em}(x,r-s)\subseteq B$ . Therefore, $(\mathord{\downarrow}\hspace{0.05em}(x,0),r)\leq^{{d_{L}}\!{^{+}}} (B,s)$ . So we conclude that $\bigvee\limits_{i\in I}^{{\bf B}(cl_{L}(\Psi(X)),d_{L})}(\mathord{\downarrow}\hspace{0.05em}(x_{i},0),r_{i})=(\mathord{\downarrow}\hspace{0.05em}(x,0),r)$ . This shows that ${\bf B}(\zeta)$ is Y-Scott continuous, and hence $\zeta$ is a Y-continuous mapping.

Proof. By Proposition 21, we have that $\hat{X}$ satisfies Condition $(\!\ast\!)$ . Obviously, $\Psi(X)\subseteq\hat{X}$ . Then $cl_{L}(\Psi(X))\subseteq\hat{X}$ . Let $A\in\hat{X}$ . By Proposition 22, we have that $(cl_{L}(\Psi(X)),d_{L})$ is a local Yoneda-complete quasi-metric space. It follows from Proposition 23 that there exists $(M,s)\in {\bf B}(cl_{L}(\Psi(X)),d_{L})$ such that $cl_{g}({\bf B}(\zeta)(A))=\mathord{\downarrow}\hspace{0.05em}(M,s)$ . Then $s=0$ and $A\subseteq M$ . Let

$$\mathcal{B}=\{(Z,t)\in B(\tilde{X},\tilde{d})\mid d_{\mathcal{H}}(Z,A)\leq t\}\cap {\bf B}(cl_{L}(\Psi(X)),d_{L}).$$

Then $\mathcal{B}$ is a g-closed subset of ${\bf B}(cl_{L}(\Psi(X)),d_{L})$ . Since ${\bf B}(\zeta)(A)\subseteq\mathcal{B}$ , we have that $cl_{g}({\bf B}(\zeta)(A))\subseteq\mathcal{B}$ . Then $(M,0)\in\mathcal{B}$ , and thus $d_{\mathcal{H}}(M,A)=0$ . So we conclude that $M\subseteq A$ . Therefore, $M=A\in cl_{L}(\Psi(X))$ . This shows that $\hat{X}=cl_{L}(\Psi(X))$ .

If direction. Let $(x_{1},r_{1}), (x_{2},r_{2})\in {\bf B}(X,d)$ satisfying $(x_{1},r_{1})\leq^{d^{+}}(x_{2},r_{2})$ . Clearly, $(f(x_{2}),r_{2})\in\mathord{\downarrow}\hspace{0.05em}(f(x_{2}),r_{2})$ , and therefore, $(x_{2},r_{2})\in({\bf B}(f))^{-1}(\mathord{\downarrow}\hspace{0.05em}(f(x_{2}),r_{2}))$ . Since $\mathord{\downarrow}\hspace{0.05em}(f(x_{2}),r_{2})$ is a g-closed set, so is $({\bf B}(f))^{-1}(\mathord{\downarrow}\hspace{0.05em}(f(x_{2}),r_{2}))$ . Thus, $(x_{1},r_{1})\in({\bf B}(f))^{-1}(\mathord{\downarrow}\hspace{0.05em}(f(x_{2}),r_{2})),$ which implies $(f(x_{1}), r_{1})\in\mathord{\downarrow}\hspace{0.05em}(f(x_{2}),r_{2})$ , whence $(f(x_{1}),r_{1})\leq^{d^{'}\!{^{+}}}(f(x_{2}),r_{2})$ .

Let $(x_{i},r_{i})_{i\in I}$ be a translational complete directed family in ${\bf B}(X,d)$ with supremum (x,r). Then $(f(x_{i}),r_{i})\leq^{d^{'}\!{^{+}}}(f(x),r)$ for any $i\in I$ . Let (y,s) be an upper bound of $(f(x_{i}),r_{i})_{i\in I}$ . Then $(f(x_{i}),r_{i})\in\mathord{\downarrow}\hspace{0.05em}(y,s)$ for any $i\in I$ , and thus $(x_{i},r_{i})\in({\bf B}(f))^{-1}(\mathord{\downarrow}\hspace{0.05em}(y,s))$ for any $i\in I$ . Obviously, $\mathord{\downarrow}\hspace{0.05em}(y,s)$ is a g-closed set. Then $({\bf B}(f))^{-1}(\mathord{\downarrow}\hspace{0.05em}(y,s))$ is a g-closed set, and thus $(x,r)\in({\bf B}(f))^{-1}(\mathord{\downarrow}\hspace{0.05em}(y,s))$ , that is, $(f(x),r)\leq^{d^{'}\!{^{+}}}(y,s)$ . Therefore, $\bigvee_{i\in I}(f(x_{i}),r_{i})=(f(x),r)$ . This implies that $(f(x_{i}),r_{i})_{i\in I}$ is a translational complete directed family in ${\bf B}(X^{'}\!,d^{'})$ . So we conclude that ${\bf B}(f)$ is a Y-Scott continuous map.

Proof. Let $(X^{'}\!,d^{'})$ be a local Yoneda-complete quasi-metric space, and $f: X\longrightarrow X^{'}$ be a Y-continuous mapping. For all A in $\hat{X}$ , there exists a unique $(y_{A},r_{A})\in {\bf B}(X^{'}\!,d^{'})$ such that $cl_{g}({\bf B}(f)(A))=\mathord{\downarrow}\hspace{0.05em}(y_{A},r_{A})$ and $r_{A}=0$ . Define a mapping $f^{\ast}: \hat{X}\longrightarrow X^{'}\!$ as follows:

$$\forall\ A\in \hat{X},\ f^{\ast}(A)=y_{A}.$$

Then $f^{\ast}$ is well-defined.

Let $x\in X$ . Then $f^{\ast}(\zeta(x))=f^{\ast}(\mathord{\downarrow}\hspace{0.05em}(x,0))$ . Since

$$cl_{g}({\bf B}(f)(\mathord{\downarrow}\hspace{0.05em}(x,0)))=cl_{g}(\{(f(y),s)\mid(y,s)\leq^{d{^{+}}}(x,0)\})=\mathord{\downarrow}\hspace{0.05em}(f(x),0),$$

we have that $f^{\ast}(\mathord{\downarrow}\hspace{0.05em}(x,0))=f(x)$ . Then $f^{\ast}\circ \zeta=f$ .

To prove this claim, we show that ${\bf B}(f^{\ast})$ is Y-Scott continuous and then use Proposition 4. Let $(B_{1},r),(B_{2},s)\in {\bf B}(\hat{X},\hat{d})$ satisfying $(B_{1},r)\leq^{\hat{d}{^{+}}}(B_{2},s)$ . Then $B_{1}+r\subseteq B_{2}+s$ , and thus $B_{1}+r-s\subseteq B_{2}$ . Therefore, ${\bf B}(f)(B_{1}+r-s)\subseteq {\bf B}(f)(B_{2})$ , and so we conclude that $cl_{g}({\bf B}(f)(B_{1}+r-s))\subseteq cl_{g}({\bf B}(f)(B_{2}))$ . Since $cl_{g}({\bf B}(f)(B_{1}))=\mathord{\downarrow}\hspace{0.05em}(y_{B_{1}},0), cl_{g}({\bf B}(f)(B_{2}))=\mathord{\downarrow}\hspace{0.05em}(y_{B_{2}},0)$ , and $cl_{g}({\bf B}(f)(B_{1}+r-s))=cl_{g}({\bf B}(f)(B_{1}))+(r-s),$ we have that $(y_{B_{1}},r-s)\leq^{d^{'}\!{^{+}}}(y_{B_{2}},0)$ . Then $d^{'}(y_{B_{1}},y_{B_{2}})\leq r-s$ . This shows that ${\bf B}(f^{\ast})(B_{1},r)\leq^{d^{'}\!{^{+}}} {\bf B}(f^{\ast})(B_{2},s)$ .

Let $(A_{i},r_{i})_{i\in I}$ be a translational complete directed family in ${\bf B}(\hat{X},\hat{d})$ satisfying $\bigvee\limits_{i\in I}^{{\bf B}(\hat{X},\hat{d})}(A_{i},r_{i})$ exists. By Proposition 21, we have that $\bigvee\limits_{i\in I}^{{\bf B}(\hat{X},\hat{d})}(A_{i},r_{i})=(cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i})),\bigwedge_{i\in I}r_{i}).$ Next, we shall prove that

$${\bf B}(f^{\ast})(\bigvee\limits_{i\in I}^{{\bf B}(\hat{X},\hat{d})}(A_{i},r_{i}))=\bigvee\limits_{i\in I}^{{\bf B}(X^{'}\!,d^{'})}{\bf B}(f^{\ast})(A_{i},r_{i}),$$

that is, $(f^{\ast}(A),\bigwedge_{i\in I}r_{i})=\bigvee\limits_{i\in I}^{{\bf B}(X^{'}\!,d^{'})}(f^{\ast}(A_{i}),r_{i}),$ where $A=cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i}))$ . Since $(A_{i},r_{i})\leq^{\hat{d}{^{+}}}(A,\bigwedge_{i\in I}r_{i})$ for any $i\in I$ , we have that $(f^{\ast}(A_{i}),r_{i})\leq^{d^{'}\!{^{+}}}(f^{\ast}(A),\bigwedge_{i\in I}r_{i})$ and $(f^{\ast}(A_{i}),r_{i})_{i\in I}$ is a directed set. Since $(X^{'}\!,d^{'})$ is a local Yoneda-complete quasi-metric space, we have that $\bigvee\limits_{i\in I}^{{\bf B}(X^{'}\!,d^{'})}{\bf B}(f^{\ast})(A_{i},r_{i})$ exists, denoted by $(y,\bigwedge_{i\in I}r_{i})$ . Thus $(y,\bigwedge_{i\in I}r_{i})\leq^{d^{'}\!{^{+}}}(f^{\ast}(A),\bigwedge_{i\in I}r_{i})$ , which implies $d^{'}\!{^{+}}(y,f^{\ast}(A))=0$ .

Since $(f^{\ast}(A_{i}),r_{i})\leq^{d^{'}\!{^{+}}} (y,\bigwedge_{i\in I}r_{i})$ , we have that $(f^{\ast}(A_{i}),r_{i}-\bigwedge_{i\in I}r_{i})\leq^{d^{'}\!{^{+}}} (y,0)$ for any $i\in I$ . From this, we may conclude

$$\begin{array}{lll}{\bf B}(f)(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i})&\subseteq cl_{g}({\bf B}(f)(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i}))\\&=cl_{g}({\bf B}(f)(A_{i}))+r_{i}-\bigwedge_{i\in I}r_{i}\\&=\mathord{\downarrow}\hspace{0.05em}(f^{\ast}(A_{i}),0)+r_{i}-\bigwedge_{i\in I}r_{i}\\&=\mathord{\downarrow}\hspace{0.05em}(f^{\ast}(A_{i}),r_{i}-\bigwedge_{i\in I}r_{i})\\&\subseteq \mathord{\downarrow}\hspace{0.05em}(y,0)\end{array}$$

for any $i\in I$ . Since ${\bf B}(f)$ is a Y-Scott continuous mapping, it follows from Proposition 26 that $({\bf B}(f))^{-1}(\mathord{\downarrow}\hspace{0.05em}(y,0))$ is a g-closed set. Hence, $A=cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i}))\subseteq({\bf B}(f))^{-1}(\mathord{\downarrow}\hspace{0.05em}(y,0))$ , and thus $cl_{g}({\bf B}(f)(A))\subseteq\mathord{\downarrow}\hspace{0.05em}(y,0)$ . By the definition of $f^{\ast}$ , this implies $\mathord{\downarrow}\hspace{0.05em}(f^{\ast}(A),0)\subseteq\mathord{\downarrow}\hspace{0.05em}(y,0)$ , and therefore, $(f^{\ast}(A),0)\leq^{{d^{'}}\!{^{+}}}(y,0)$ , which implies $d^{'}\!{^{+}}(f^{\ast}(A),y)=0$ . Together with $d^{'}\!{^{+}}(y,f^{\ast}(A))=0$ from above, $f^{\ast}(A)=y$ follows. Therefore, $\bigvee\limits_{i\in I}^{{\bf B}(X^{'}\!,d^{'})}(f^{\ast}(A_{i}),r_{i})=(f^{\ast}(A),\bigwedge_{i\in I}r_{i})$ . Hence $f^{\ast}$ is a Y-continuous mapping.

Suppose that there exists a Y-continuous mapping $g:\hat{X}\longrightarrow X^{'}\!$ such that $g\circ\zeta=f$ . Let $\mathcal{C}=\{A\in\hat{X}\mid f^{\ast}(A)=g(A)\}$ . Then $\mathcal{C}\subseteq \hat{X}\subseteq\tilde{X}$ and $\Psi(X)\subseteq \mathcal{C}$ . Next, we shall prove that $\mathcal{C}$ satisfies the Condition $(\!\ast\!)$ . Let $(A_{i},r_{i})_{i\in I}$ be a directed family in ${\bf B}(\mathcal{C},d_{\mathcal{H}}\mid_{\mathcal{C}\times \mathcal{C}})$ with an upper bound (B,s). By Propositions 20 and 21, we have that $\bigvee\limits_{i\in I}^{{\bf B}(\tilde{X},\tilde{d})}(A_{i},r_{i})=(A,\bigwedge_{i\in I}r_{i})$ and $A\in\hat{X}$ , where $A=cl_{g}(\bigcup_{i\in I}(A_{i}+r_{i}-\bigwedge_{i\in I}r_{i}))$ . Obviously, $(A_{i},r_{i}-\bigwedge_{i\in I}r_{i})_{i\in I,\sqsubseteq}$ is a Cauchy weighted net in ${\bf B}(\hat{X},\hat{d})$ . Let $Z\in \hat{X}$ . Then, $\hat{d}(A_{i},Z)\leq\hat{d}(A_{i},A)+\hat{d}(A,Z)\leq r_{i}-\bigwedge_{i\in I}r_{i}+\hat{d}(A,Z)$ for any $i\in I$ , and thus $\hat{d}(A_{i},Z)-r_{i}+\bigwedge_{i\in I}r_{i}\leq\hat{d}(A,Z)$ for any $i\in I$ . Let s be an upper bound of $(\hat{d}(A_{i},Z)-r_{i}+\bigwedge_{i\in I}r_{i})_{i\in I}$ . If $s=+\infty$ , then $\hat{d}(A,Z)\leq s$ . If $s<+\infty$ , then $\hat{d}(A_{i},Z)-r_{i}+\bigwedge_{i\in I}r_{i}\leq s$ for any $i\in I$ , and thus $\hat{d}(A_{i},Z)\leq s+r_{i}-\bigwedge_{i\in I}r_{i}$ for any $i\in I$ . Hence, $(A_{i},r_{i}+s)\leq^{\hat{d}{^{+}}}(Z,\bigwedge_{i\in I}r_{i})$ for any $i\in I$ , and so we conclude that $(A,\bigwedge_{i\in I}r_{i}+s)\leq^{\hat{d}{^{+}}}(Z,\bigwedge_{i\in I}r_{i})$ . This implies $\hat{d}(A,Z)\leq s$ . Therefore, $\bigvee_{i\in I}(\hat{d}(A_{i},Z)-r_{i}+\bigwedge_{i\in I}r_{i})=\hat{d}(A,Z)$ . By Lemma 2.6 (i) of Ng and Ho (Reference Ng and Ho2017), A is the d-limit of $(A_{i})_{i\in I}$ . Since $f^{\ast}$ and g are Y-continuous, we have that $f^{\ast}(A)=f^{\ast}(\lim_{d}A_{i})=\lim_{d}f^{\ast}(A_{i})=\lim_{d}g(A_{i})=g(\lim_{d}A_{i})=g(A).$ So we conclude $A\in \mathcal{C}$ . It follows from Proposition 24 that $\hat{X}=cl_{L}(\Psi(X))=\mathcal{C}$ , and therefore $f^{\ast}=g$ .

Consequently, $(\hat{X},\hat{d})$ is the local Yoneda completion of (X,d).