Nonvanishing derived limits in shape theory (Q1913485)

It is known that for a compact Hausdorff space \(X\) the \(n\)th derived limit of the \(m\)th homology progroup vanishes, for all \(n \geq 2\) and \(m \geq 1\). The author constructs paracompact spaces \(X\), for which the groups \(\lim^n\text{pro-} H_m(X;Z)\) do not vanish. For this, by using results on progroups by Mitchell and Grobbot, he first produces a simple abelian progroup \(P^n\) such that \(\lim^nP^n \neq 0\), and then for every integer \(m\geq 1\) he constructs an inverse system of polyhedra such that its \(m\)th homology progroup equals \(P^n\). This inverse limit is a paracompact space with \(\lim^n \text{pro-} H_m(x) \neq 0\).