Apart from simply-connected spaces, a non simply-connected co-H-space is a
typical example of a space X with a co-action of BΟ1β(X) along rX:XβBΟ1β(X) the classifying map of the universal covering. If such
a space X is actually a co-H-space, then the fibrewise p-localization of rX
(or the `almost' p-localization of X) is a fibrewise co-H-space (or an `almost'
co-H-space, resp.) for every prime p. In this paper, we show that the converse
statement is true, i.e., for a non simply-connected space X with a co-action of
BΟ1β(X) along rX, X is a co-H-space if, for every prime p, the almost
p-localization of X is an almost co-H-space.Comment: 10 pages, no figure