Every finite system of T1 uniformities comes from a single distance structure

Using the general notion of distance function introduced in [2], a construction of the finest distance structure (d, M, P ) that induces a given quasiuniformity is given, which leads to a concrete, full, and co-reflective embedding of the category of quasi-uniformities into that of distance spaces. Moreover, when the usual defining condition xUεy :⇔ d(y, x) 6 ε of the basic entourages is generalized to nd(y, x) 6 nε (for a fixed positive integer n), it turns out that if the value-monoid M is commutative, one gets a countably infinite family (Un)n∈ω of quasi-uniformities on X. It is then shown that at least every finite system and every descending chain of T1 quasi-uniformities that fulfill a weak symmetry condition is included in such a system. This is only possible since, in contrast to real metric spaces, d need not be symmetric.