On dimensions modulo a compact metric ANR and modulo a simplicial complex

Abstract

V. V. Fedorchuk has recently introduced dimension functions K-dim \leq K-Ind and L-dim \leq L-Ind, where K is a simplicial complex and L is a compact metric ANR. For each complex K with a non-contractible join |K| * |K| (we write |K| for the geometric realisation of K), he has constructed first countable, separable compact spaces with K-dim < K-Ind. In a recent paper we have combined an old construction by P. Vop\v{e}nka with a new construction by V. A. Chatyrko, and have assigned a certain compact space Z (X, Y) to any pair of non-empty compact spaces X, Y. In this paper we investigate the behaviour of the four dimensions under the operation Z (X, Y). This enables us to construct more examples of compact Fr\'echet spaces which have prescribed values K-dim < K-Ind, L-dim < L-Ind, or K-Ind < |K|-Ind, and (connected) components of which are metrisable.