A subsequence theorem for generalised inverse limits

In this paper we give a subsequence theorem for generalised inverse limits. Suppose that for each i∈Ni∈N, XiXi is a compact Hausdorff space and fi+1:Xi+1→Xifi+1:Xi+1→Xi is a continuous function. The subsequence theorem for inverse limits states that if 〈si:i∈N〉〈si:i∈N〉 is a strictly increasing sequence of positive integers where s0=1s0=1, and if Y0=X0Y0=X0, for each positive integer i  , Yi=XsiYi=Xsi, and for each i∈Ni∈N, gi+1=fsi∘⋯∘fsi+1gi+1=fsi∘⋯∘fsi+1, then lim←(Xi,fi)=lim←(Yi,gi). The subsequence theorem does not hold for generalised inverse limits. We give a class of upper semicontinuous set-valued bonding functions for which the theorem does hold.