On cell-like mappings and the dimensional full-valuedness of compacta (Q1907413)

In dimension theory examples are well known of dimensionally non-full-valued compacta, the dimension of the product of which being smaller than the sum of the dimensions of the factors. Recently, \textit{A. N. Dranishnikov} [Sov. Math., Dokl. 37, No. 3, 769-773 (1988); translation from Dokl. Akad. Nauk SSSR 300, No. 5, 1045-1049 (1988; Zbl 0669.54020)] constructed two four-dimensional ANR-compacta with semi-dimensional product, proving the existence of dimensionally non-full-valued ANR-compacta. Then \textit{V. L. Sukhikh} [Mosc. Univ. Math. Bull. 48, No. 2, 56-57 (1993); translation from Vestn. Mosk. Univ., Ser. I 48, No. 2, 104-106 (1993; Zbl 0809.54013)] has shown that any ANR-compactum can be represented in form of the cell-like image of a certain dimensionally full-valued ANR-compactum of the same dimension. The objective of this paper is to show that any compactum is the cell-like image of a dimensionally full-valued compactum with the same dimension. As dimensionally full-valued compacta we take \(\Delta\)-compacta, by which we mean compacta that can be represented in the form of the inverse limit of an inverse sequence of simplicial complexes and simplicial mappings [\textit{Y. Kodama}, Fundam. Math. 89, 13-22 (1975; Zbl 0314.55011)]. In addition, the proved theorem sharpens the theorem of \textit{R. D. Edwards} [\textit{J. J. Walsh}, Lect. Notes Math. 870, 105-118 (1981; Zbl 0474.55002)], using some constructions of the author [Russ. Math. Surv. 47, No. 4, 214-215 (1992); translation from Usp. Mat. Nauk 47, No. 4(286), 197-198 (1992; Zbl 0784.54036)].