Is the shape theory of Hausdorff compact spaces complete with respect to the inverse limit operation?--complete means that for every inverse system of Hausdorff compact spaces, and the shape morphisms between them, there exists a Hausdorff compact space which is the (inverse) limit (in the shape category) of the system. Remember that the shape morphisms do not have to be--and in general they are not--induced by continuous maps.
WARNING: the shape theory of Hausdorff compact spaces is continuous with respect to inverse limit; this (by itself) doesn't mean that the shape theory of Hausdorff compact spaces is complete--as long as I know, it is as open question as it was since the time when the shape theory of Hausdorff compact spaces was defined.
REMARK: the inverse limit of a shape inverse sequence of Hausdorff compact spaces always exists; the countable case is fine.