J.H. Michael recently proved that there exists a metric semigroup
Usuch that every compact metric semigroup with property Pis topologically isomorphic to a subsemigroup of U; where a semigroup Shas property Pif and only if for each x, yin S, x≠ y, there is a zin Ssuch that xs≠ yzor zx≠ zy
A stronger result is proved here more simply. It is shown that for any set
Aof metric semigroups there exists a metric semigroup Usuch that each Sin Ais topologically isomorphic to a subsemigroup of U. In particular this is the case when Ais the class of all separable metric semigroups, or more particularly the class of all compact metric semigroups.