A Note on Quasi-metric Spaces

In the present note it is shown that a very useful generalized distance function may be defined in certain of these spaces. Clearly, any such distance function must be an asymmetric one. W. A. Wilson considered the definition of asymmetric distances in certain spaces which satisfy stronger separation axioms than K. I t is shown here that a slight modification of one of the axioms in [W] allows the extension of a large part of the theory developed there to spaces subject to K. Since many of the theorems and proofs in [W] remain valid here with only very obvious changes, this note will be limited to a mere sketch concerning new properties which arise under the weaker axioms used. The reader should have no difficulty in adapting the more complete discussion given in [W] to the case studied here.