A study on quasi-pseudometrics

Abstract

We study some aspects of the space $QPM(X)$ of all quasi-pseudometrics on a set $X$ equipped with the extended $T_0$-quasi-metric $A_X(f,g)=\sup_{(x,y)\in X\times X}(f(x,y)-g(x,y))$ whenever $f,g\in QPM(X)$. We observe that this space is bicomplete and exhibit various closedsubspaces of $( QPM(X), \tau((A_X)^s))$.In the second part of the paper, as a rough way to measure the asymmety of a quasi-pseudometric $f$ on a set $X$, we investigate some properties of the value $(A_X)^s(f,f^{-1}).