Some comments on $\tau$-distance and existence theorems in complete metric spaces

Abstract

Very recently, we have introduced the concept of $\tau'$-distance,
 which is slightly weaker than that of $\tau$-distance.
We discuss the difference between both concepts,
 proving some existence theorems in complete metric spaces and
 giving an example of a $\tau'$-distance which is not a $\tau$-distance.