One of the difficult problems in extension theory is the question of whether there exist universal metrizable compacta within the class of metrizable compacta that are absolute co-extensors for a given CW-complex K. In particular, for a nontrivial abelian group G and $ngeq 2$, this problem specializes as follows. Does there exist a universal metrizable compactum X so that $dim _GXleq n$? By $dim _GX$ is meant the cohomological dimension of X with respect to the abelian group G which can be defined in terms of extension theory. The meaning here of ''universal'' is that for each compact metrizable space Y with $dim _GYleq n$, there exists an embedding of Y in X.

Recent developments indicate that methods using direct systems of spaces and their limits can be useful in resolving this question. We will explain what is meant by a direct system and its limit and will speak about some of the properties of such systems that will be needed if they are to be applied in this part of extension theory.