Topics in dimension theory (Q2874862)

This is a good survey which outlines those parts of dimension theory where important developments occurred under the last 10--15 years. Interesting results mentioned in the survey explain the authors' choice of topics. Open problems placed between the statements indicate some main directions of future studies. A huge list of references (184) is ready to help readers with details.NEWLINENEWLINEExcept for the introduction and preliminaries the survey consists of 6 main parts: Cohomological dimension and extension theories, Pasynkov-Torunczyk theorem and related topics, Universal spaces and related topics, Topics in infinite dimension theory, Miscellaneous topics and Asymptotic dimension.NEWLINENEWLINEIn the first part the authors recall several recent acyclic resolution theorems, describe the relationship between cohomological and extension dimensions, present Menger-Urysohn and Hurewicz formulae in terms of extension dimension and the dimension of products.NEWLINENEWLINEIn the second part they consider different extension dimension versions of some known results by Pasynkov and Torunczyk about functions between metrizable compacta with finite-dimensional ranges and pre-images of points.NEWLINENEWLINEIn the third part the authors discuss different topics related to universality, in particular: characterization theorems for finite-dimensional Polish spaces to be Nöbeling manifolds, properties of Erdös space and complete Erdös space, universal spaces for metric spaces of weight \(\tau\).NEWLINENEWLINEIn the fourth part they mention some statements related to \(C\)-spaces and weakly infinite-dimensional spaces, in particular, their connection to selection theorems. A short discussion on two transfinite extensions of \(dim\) in these families of spaces is also presented as well as mentioning families of Cantor manifolds without weakly infinite-dimensional subcontinua which have some effect to Hendersons compacta.NEWLINENEWLINEIn the fifth part the authors recall interesting results of different type. Let us list only the names of topics: Pontrjagin-Schnirelmann theorem, embeddings of \(n\)-dimensional compacta into products of \(1\)-dimensional compacta, Dimensionsgrad-Brouwer dimension, Compactness degree and compact deficiency, Weakly \(n\)-dimensional spaces, \(n\)-points sets, Coloring of mappings, The gap between the dimensions, The Menger-Urysohn formula for \(ind\).NEWLINENEWLINEThe sixth part is devoted to asymptotic dimension. Here one can find a short background with some definitions and a list of intricated theorems.NEWLINENEWLINEA remark to a sentence on page 383: in [\textit{V. V. Fedorchuk} and \textit{V. A. Chatyrko}, Vestn. Mosk. Univ., Ser. I 2004, No. 6, 50--52 (2004); translation in Mosc. Univ. Math. Bull. 59, No. 6, 34--36 (2004; Zbl 1084.54006)] it was proved that \(cmp Z_3 = 2\).NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].