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

TL;DR

This paper explores the behavior of certain dimension functions in compact spaces related to simplicial complexes and ANRs, constructing examples with prescribed dimension inequalities and analyzing their properties.

Contribution

It investigates the behavior of dimension functions under a new operation Z(X,Y), enabling the construction of compact spaces with specific dimension inequalities.

Findings

Constructed compact Fréchet spaces with prescribed dimension inequalities.

Analyzed the behavior of K-dim, K-Ind, L-dim, and L-Ind under the operation Z(X,Y).

Provided new examples of spaces with connected components that are metrizable.

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.