Charalambous, M. G. Universal spaces for locally finite-dimensional Tychonoff spaces. (English) Zbl 0675.54035 Proc. Am. Math. Soc. 106, No. 2, 507-514 (1989). Covering dimension dim is defined in terms of cozero covers and all spaces are assumed to be at least Tychonoff. A space is called locally finite-dimensional if it has an open cover consisting of finite- dimensional sets. One may replace in this definition cozero for open. The space is said to be strongly locally finite-dimensional if in addition the cover is countable. The author gives us results such as: 1. The class of all locally finite- dimensional spaces of weight \(\leq \tau\) has a universal element that is locally compact. 2. The class of all strongly locally finite-dimensional spaces of weight \(\leq \tau\) has a universal element that is \(\sigma\)- compact and locally compact. 3. The class of all spaces with loc dim\(\leq n\) and weight \(\leq \tau\) has a universal element Z which is compact with dim \(Z\leq n.\) Suppose f: \(X\to Y\) is continuous. Then W(f) is meant the smallest cardinal \(\alpha\) for which there exists a space Z of weight \(\alpha\) and an embedding g: \(X\to Y\times Z\) with \(f=proj_ Y\circ g\). Along these lines, the author obtains results similar to those above, a sample of which is: 4. The class of all locally finite-dimensional spaces that can be mapped by some continuous function f with W(f)\(\leq \tau\) into a metrizable space of weight \(\theta\) has a universal element which is locally Čech-complete and paracompact. Reviewer: L.Rubin Cited in 1 Document MSC: 54F45 Dimension theory in general topology 54E15 Uniform structures and generalizations Keywords:local dimension; locally Čech-complete paracompact universal element; Covering dimension; locally finite-dimensional spaces; weight; strongly locally finite-dimensional spaces PDFBibTeX XMLCite \textit{M. G. Charalambous}, Proc. Am. Math. Soc. 106, No. 2, 507--514 (1989; Zbl 0675.54035) Full Text: DOI References: [1] A. Arhangel’ skiĭ, Factorization of mappings according to weight and dimension, Soviet Math. Dokl. 8 (1967), 731-734. · Zbl 0174.54202 [2] M. G. Charalambous, A new covering dimension function for uniform spaces. part 2, J. London Math. Soc. (2) 11 (1975), no. part 2, 137 – 143. · Zbl 0306.54048 [3] -, Further theory and applications of covering dimension of uniform spaces (to appear). · Zbl 0776.54024 [4] C. H. Dowker, Local dimension of normal spaces, Quart. J. Math. Oxford Ser. (2) 6 (1955), 101 – 120. · Zbl 0066.41204 [5] Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna. Tom 47. [Mathematics Library. Vol. 47]. Ryszard Engelking, General topology, PWN — Polish Scientific Publishers, Warsaw, 1977. Translated from the Polish by the author; Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. [6] Ryszard Engelking, Teoria wymiaru, Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51]. Ryszard Engelking, Dimension theory, North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN — Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author; North-Holland Mathematical Library, 19. [7] Leonid Luxemburg, On universal metric locally finite-dimensional spaces, General Topology Appl. 10 (1979), no. 3, 283 – 290. · Zbl 0413.54043 [8] B. A. Pasynkov, On universal bicompacta of a given weight and dimension, Soviet Math. Dokl. 5 (1964), 245-246. · Zbl 0197.48601 [9] -, A factorization theorem for non-closed sets, Soviet Math. Dokl. 13 (1972), 292-295. · Zbl 0247.54037 [10] B. A. Pasynkov, Factorization theorems in dimension theory, Uspekhi Mat. Nauk 36 (1981), no. 3(219), 147 – 175, 256 (Russian). · Zbl 0477.54021 [11] B. A. Pasynkov, On dimension theory, Aspects of topology, London Math. Soc. Lecture Note Ser., vol. 93, Cambridge Univ. Press, Cambridge, 1985, pp. 227 – 250. [12] A. R. Pears, Dimension theory of general spaces, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975. · Zbl 0312.54001 [13] B. R. Wenner, A universal separable metric locally finite-dimensional space, Fund. Math. 80 (1973), no. 3, 283 – 286. · Zbl 0265.54030 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.