Fedorchuk, V. V. The Urysohn identity and dimension of manifolds. (English. Russian original) Zbl 0967.54033 Russ. Math. Surv. 53, No. 5, 937-974 (1998); translation from Usp. Mat. Nauk. 53, No. 5, 73-114 (1998). In this beautiful survey article the author presents a brief history and all the details about the contents of the Uryson identity as well as the dimension of manifolds. In the first section he discusses the Uryson identity. There are three well-known dimension functions viz. ind, the small inductive dimension, Ind, the large inductive dimension and dim, the Lebesgue covering dimension for a space \(X\). For a compact metric space \(X\) it was proved by Uryson himself that \(\text{ind} X=\text{Ind} X=\dim X\), and this is now famous as the Uryson identity. The basic question dealt with in this article is to examine as to how much general a space \(X\) can be taken so that the Uryson identity is valid for \(X\). According to the author, this question has been constantly studied by several dimension-theorists who have contributed significantly in proving the identity for larger and larger classes of spaces or by providing counterexamples. The research went in both directions. In this respect, the counterexample discovered by the Indian mathematician Prabir Roy of a complete metric space \(X\) for which \(\text{ind} X=0\) but \(\text{Ind} X=1\) is also mentioned. While discussing various improvements, the author raises very interesting open problems of the area of research and this is characterestic of all survey articles published in Russian Mathematical Surveys.In the second section the author first discusses the topology of non-metrizable manifolds. These spaces have recently attracted the attention of several topologists also interested in the foundations of mathematics viz. set theory. Spaces like the Long line are interesting examples of nonmetrizable manifolds. The first manifold with non-coinciding dimensions was constructed by the author jointly with Fillipov. Again several interesting questions have been identified and discussed. At the end of the article the author presents the results related to the cohomological dimension of a space. The most interesting aspect of this dimension concerns Aleksandrov’s problem which has now been solved by A. N. Dranishnikov. The author briefly touches upon various results related to the work of Dranishnikov and its other extensions. Reviewer: Satya Deo (Jabalpur) Cited in 3 Documents MSC: 54F45 Dimension theory in general topology 55M10 Dimension theory in algebraic topology Keywords:Uryson identity; cohomological dimension × Cite Format Result Cite Review PDF Full Text: DOI