DIFFERENT TYPES OF QUASI-METRIC AND PARTIAL METRIC SPACES

Abstract

The notion of a partial metric space was introduced by S. Matthews \cite{Matthews1992} in 1992. This notion arose as a certain extension of the notion of metric spaces and was used in computer science, where there are non-Hausdorff topological models. A function $p:X^2\to [0,+\infty)$ is called {\it a partial metric} on $X$ if for all $x,y,z\in X$ the following conditions hold: $(p_1)$ $x=y$ if and only if $p(x,x)=p(x,y)=p(y,y)$; $(p_2)$ $p(x,x)\leq p(x,y)$; $(p_3)$ $p(x,y)=p(y,x)$; \mbox{$(p_4)$ $p(x,z)\leq p(x,y)+p(y,z)-p(y,y)$.} The topology of a partial metric space $(X,p)$ is generated by the corresponding quasi-metric $q_p(x,y)=p(x,y)-p(x,x)$. Topological and metrical properties of partial metric spaces have been studied by many mathematicians. According to \cite{HWZ}, a quasi-metric space $(X,q)$ is called: {\it sequentially isosceles} if $\lim\limits_{n\to\infty}q(y,x_n)=q(y,x)$ for any $y\in X$ and every sequence of $x_n\in X$ that converges to $x\in X$; {\it sequentially equilateral} if a sequence of $y_n\in X$ converges to $x\in X$ while there exists a convergent to $x$ sequence of $x_n\in X$ with $\lim\limits_{n\to\infty}q(y_n,x_n)=0$; {\it sequentially symmetric} a sequence of $x_n\in X$ converges to $x\in X$ while $\lim\limits_{n\to\infty}q(x_n,x)=0$; {\it metric-like} if $\lim\limits_{n\to\infty}q(x_n,x)=0$ for every convergent to $x\in X$ sequence of $x_n\in X$. It was proved in \cite{HWZ} and \cite{Lu-2020} that: $(i)$ every sequentially equilateral quasi-metric space is sequentially symmetric; $(ii)$ every metric-like quasi-metric space is sequentially isosceles; $(iii)$ every metric-like and sequentially symmetric quasi-metric space is sequentially equilateral. A topological characterization of sequentially isosceles, sequentially equilateral, sequentially symmetric and metric-like quasi-metric spaces were obtained. Moreover, examples which show that there are no other connections between the indicated types of spaces, except for $(i)-(iii)$ even in the class of metrizable partial metric spaces have been constructed.