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On the dimensional properties of Stone-Čech remainder of $ P\sb 0$-spaces


Author: H. Attia
Journal: Proc. Amer. Math. Soc. 121 (1994), 1245-1249
MSC: Primary 54F45; Secondary 54D40, 54E18
DOI: https://doi.org/10.1090/S0002-9939-1994-1156462-5
MathSciNet review: 1156462
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Abstract: A space X is called a $ {P_0}$-space if there exists a perfect mapping f from X onto a metric space Y such that $ \dim f = \sup \{ {f^{ - 1}}(y):y \in Y\} = 0$. We prove that the $ {P_0}$-space X is almost weakly infinite dimensional iff the remainder $ \beta X\backslash X$ of the Stone-Čech compactification $ \beta X$ of X is A-weakly infinite dimensional. Furthermore we prove that $ \Delta (\beta X\backslash X = {\text{ind}}(\beta X\backslash X) = {\text{Ind}}(\beta X\backslash X) = \dim (\beta X\backslash X)$ for the $ {P_0}$-space X.


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  • [1] P. S. Alexcandroff and B. A. Pasynkov, Introduction to dimension theory, Nauka, Moscow, 1973. (Russian)
  • [2] A. Arhangel′skiĭ, The factorization of mappings by weight and dimension, Dokl. Akad. Nauk SSSR 174 (1967), 1243–1246 (Russian). MR 0216472
  • [3] A. V. Arhangel′skiĭ, A class of spaces which contains all metric and all locally compact spaces, Mat. Sb. (N.S.) 67 (109) (1965), 55–88 (Russian). MR 0190889
  • [4] A. V. Arhangel′skiĭ, Open and close-to-open mappings. Relations among spaces, Trudy Moskov. Mat. Obšč. 15 (1966), 181–223 (Russian). MR 0206909
  • [5] M. M. Čoban, Topological structure of subsets of topological groups and their factor spaces, Cb. Mat. Recearsh Keshenev, Science, 44 (1977), 117-163. (Russian)
  • [6] M. M. Čoban and H. Attia, On the dimension of remainders of extensions of normal spaces, Topology Appl. 36 (1990), no. 2, 97–109. Seminar on General Topology and Topological Algebra (Moscow, 1988/1989). MR 1068163, https://doi.org/10.1016/0166-8641(90)90002-J
  • [7] R. Engleking, Dimension theory, Warszawa, Oxford and New York, 1978.
  • [8] Melvin Henriksen and J. R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1957), 83–105. MR 0096196
  • [9] Stoyan Ĭ. Nedev, Selection and factorization theorems for set-valued mappings, Serdica 6 (1980), no. 4, 291–317 (1981). MR 644284
  • [10] B. A. Pasynkov, Factorization theorems in dimension theory, Uspekhi Mat. Nauk 36 (1981), no. 3(219), 147–175, 256 (Russian). MR 622723
  • [11] B. A. Pasynkov, The dimension of spaces with a bicompact transformation group, Uspehi Mat. Nauk 31 (1976), no. 5(191), 112–120 (Russian). MR 0445470
  • [12] V. I. Ponomarev, Paracompacta: their projection spectra and continuous mappings, Mat. Sb. (N.S.) 60 (102) (1963), 89–119 (Russian). MR 0152980

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1156462-5
Keywords: $ {P_0}$-spaces, dimension, almost weakly infinite dimensional, remainder
Article copyright: © Copyright 1994 American Mathematical Society