Extensions of the inverse limit

Extensions of inverse limit space is investigated. Let P be a topological property. We say that a space Y is a P-extension of a topological space X if Y has the property P, and that there exists a subspace Z of Y such that X and Z are homeomorphic and Z is dense in Y. Let X = [X_a, f_(ab), A}; be an inverse system. In the present paper we investigate the relationship between the P-extension of the space lim X and the P-extension of the spaces X_a, aeA.