Pasynkov, B. A. On dimension theory. (English) Zbl 0573.54029 Aspects of topology, Mem. H. Dowker, Lond. Math. Soc. Lect. Note Ser. 93, 227-250 (1985). [For the entire collection see Zbl 0546.00024.] There are two parts. The first contains proofs of results stated in Dokl. Akad. Nauk SSSR 267, 654-658 (1982; Zbl 0561.54028)], while the second is strictly a discussion of recent discoveries of Soviet topologists in dimension theory. The spaces considered are topological spaces and the dimension function dim is defined in terms of refining finite functionally open covers by covers of the same type and of suitable order. In part I, two inequalities are considered: \[ (*)\quad \dim A\leq \dim X \text{ for }A\subset X, \qquad (**)\quad \dim (X\times Y)\leq \dim X+\dim Y. \] A sufficient condition is provided for (\(*\)) to be true, that is, if the subspace A satisfies a condition called d-right. As a corollary, it is shown that if A is completely paracompact and X is completely regular at all points of A, then (\(*\)) holds because A is d-right. The theory is applied to topological groups to show that loc dim G\(=\dim G\). As for (\(**\)), the term piecewise rectangular, for a topological product, is introduced. It is proved that (\(**\)) is true if the product is piecewise rectangular. Then the author characterizes, for Tychonoff spaces, when a product is piecewise rectangular using the notion d-right. The term piecewise rectangular is also used to designate certain inverse systems of spaces, and it is determined that in such a system if all the spaces have dim\(\leq n\), then their limit has dim\(\leq n.\) Part II contains information about the relations among the dimension invariants: covering dimension dim, small inductive dimension ind, and large inductive dimension Ind. Classes of spaces are considered. For example, chainable bicompacta, homogeneous bicompacta, algebraically homogeneous spaces. There is a discussion about examples which show that dim, ind, Ind do not necessarily agree, and on the other hand, in some cases even all three agree. The behavior of the same dimensional invariant on different, but in some sense equivalent spaces is considered. In particular so-called M-equivalent and \(\ell\)-equivalent spaces are examined. It is noted that under certain conditions ”equivalent” spaces have the same dimension dim. These kinds of results have also been extended to cohomological dimension, \(\dim_ G\). There is a discussion of infinite dimensional spaces. Spaces that are locally finite dimensional are considered, and it is stated that there are results about the existence of universal spaces in this class. Also there is a discussion of ind and the existence for each ordinal \(\alpha\) of a strongly paracompact metrizable space with \(ind=\alpha\). A dimension invariant \(Ind^+\) is introduced, which is an extension of Ind over a wider class of metrizable spaces. Finally, some results about superparacompacta are stated, and a result about bicompacta which are the continuous images of zero-dimensional homogeneous bicompacta is stated. There is a large set of questions, which indicate directions for future research in the field. Reviewer: L.Rubin Cited in 7 Documents MSC: 54F45 Dimension theory in general topology 55M10 Dimension theory in algebraic topology 54-02 Research exposition (monographs, survey articles) pertaining to general topology Keywords:problems; dimension inequalities; piecewise rectangular product; topological groups; covering dimension; small inductive dimension; large inductive dimension; chainable bicompacta; homogeneous bicompacta; algebraically homogeneous spaces; dimensional invariant; equivalent spaces; cohomological dimension; infinite dimensional spaces; universal spaces; strongly paracompact metrizable space; superparacompacta; images of zero-dimensional homogeneous bicompacta Citations:Zbl 0546.00024; Zbl 0561.54028 PDFBibTeX XML