  This paper investigates Smyth completeness of categories enriched over a
quantale obtained by equipping the unit interval of real numbers with a
continuous t-norm. A real-enriched category is Smyth-complete if each of its
forward Cauchy nets has a unique limit in the open ball topology of its
symmetrization. It is demonstrated that Smyth completeness can be characterized
as a categorical property and as a real-valued topological property.
Explicitly, it is shown that a real-enriched category is Smyth complete if and
only if it is separated and all of its ideals are representable, if and only if
its Alexandroff real-valued topology is sober.
