Quasi-metric spaces, by Salvador Romaguera

Topology Atlas Document # taic-12.htm | Production Editor: R. Flagg

Topology Atlas Invited Contributions, vol. 1, issue 5 (1996), 83.

Quasi-metric spaces
by

Salvador Romaguera

(Universidad Polit\'ecnica de Valencia, 46071 Valencia, Spain)

A quasi-pseudometric on a set $X$ is a nonnegative real valued function $d$ on $X$ times $X$ such that for all $x,y,z\in X$:

(i) $d(x,x)=0$ and

(ii) $d(x,y)<= d(x,z)+d(z,y)$ .

If $d$ satisfies the additional condition

(iii) $d(x,y)=0$ if and only if $x=y$,

then $d$ is called a quasi-metric on $X$.

A quasi-(pseudo)metric space is a pair $(X,d)$ such that $X$ is a (nonemtpy) set and $d$ is a quasi-(pseudo)metric on $X$. A topological space $(X,T)$ is called quasi-(pseudo)metrizable if there is a quasi-(pseudo)metric on $X$ compatible with $T$.

Quasi-metric spaces were introduced by A.W.Wilson (Amer. J. Math., 53(1931), 675-684) for purely topological reasons. However, several generalized distance functions were already used by E.W.Chittenden (1927), V.W.Niemytzki (1927) and others, in their study of the metrization problem.

In a paper published in 1963, J.C.Kelly began a systematized study of quasi-metrizable spaces from a bitopological point of view. Thus it appears in a natural way the problem of characterizing both topological and bitopological spaces that admit a compatible quasi-metric in a similar way to the celebrated Nagata-Smirnov metrization theorem. In this direction several contributions have been obtained (S.Salbany (1972), R.Fox (unpublished), R.D.Kopperman (1993), etc.) but the cited problem is still open.

In the last years quasi-pseudometric spaces and, particulary, complete quasi-pseudometric spaces have been an appropriate tool in theoretical computer science.

In my opinion the following references are basic for the people interested in quasi-metric spaces.

(1) J.Kofner, On quasi-metrizability, Topology Proc., 5(1980), 111-138.

(2) P.Fletcher and W.F.Lindgren, Quasi-Uniform Spaces, Marcel Dekker, New York, 1982.

(3) R.Fox, Solution of the $\gamma $space problem, Proc. Amer. Math. Soc, 85(1982), 606-608.

(4) H.P.A.K\"unzi, Complete quasi-pseudo-metric spaces, Acta Math. Hungar., 59(1992), 121-146.

(5) H.P.A.K\"unzi, Quasi-uniform spaces - eleven years later. Topology Proc., 18(1993), 143-171.

Received by the editors: February 9, 1996.