Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An internal characterization of paracompact $ p$-spaces


Author: R. A. Stoltenberg
Journal: Trans. Amer. Math. Soc. 196 (1974), 249-263
MSC: Primary 54D20
DOI: https://doi.org/10.1090/S0002-9947-1974-0346746-4
MathSciNet review: 0346746
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to characterize paracompact p-spaces in terms of spaces with refining sequences $ \bmod\;k$. A space X has a refining sequence $ \bmod\;k$ if there exists a sequence $ \{ {\mathcal{G}_n}\vert n \in N\} $ of open covers for X such that $ \cap _{n = 1}^\infty {\text{St}}(C,{\mathcal{G}_n}) = P_C^1$ is compact for each compact subset C of X and $ {\text{\{ St}}{(C,{\mathcal{G}_n})^ - }\vert n \in N\} $ is a neighborhood base for $ P_C^1$. If $ P_C^1 = C$ for each compact subset C of X then X is metrizable. On the other hand if we restrict the set C to the family of finite subsets of X in the above definition then we have a characterization for strict p-spaces. Moreover, in this case, if $ P_C^1 = C$ for all such sets then X is developable. Thus the concept of a refining sequence $ \bmod\;k$ is natural and it is helpful in understanding paracompact p-spaces.


References [Enhancements On Off] (What's this?)

  • [1] P. S. Aleksandrov, On some results concerning topological spaces and their continuous mappings, General Topology and its Relations to Modern Analysis and Algebra (Proc. Sympos., Prague, 1961), Academic Press, New York; Publ. House Czech. Acad. Sci. Prague, 1962, pp. 41-54. MR 26 #3003. MR 0145472 (26:3003)
  • [2] A. V. Arhangel'skiĭ, On a class of spaces containing all metric and all locally bicompact spaces, Dokl. Akad. Nauk SSSR 151 (1963), 751-754 = Soviet Math. Dokl. 4 (1963), 1051-1055. MR 27 #2959. MR 0152988 (27:2959)
  • [3] -, Bicompact sets and the topology of spaces, Trudy Moskov Mat. Obšč. 13 (1965), 3-55 = Trans. Moscow Math. Soc. 1965, 1-62. MR 33 #3251. MR 0195046 (33:3251)
  • [4] -, On a class of spaces containing all metric spaces and all locally compact spaces, Mat. Sb. 67 (109) (1965), 55-88; English transl., Amer. Math. Soc. Transl. (2) 92 (1970), 1-39. MR 32 #8299; 42 #3. MR 0190889 (32:8299)
  • [5] -, Mappings and spaces, Uspehi Mat. Nauk 21 (1966), no. 4(130), 133-184 = Russian Math. Surveys 21 (1966), no. 4, 115-162. MR 37 #3534. MR 0227950 (37:3534)
  • [6] D. K. Burke, On subparacompact spaces, Proc. Amer. Math. Soc. 23 (1969), 655-663. MR 40 #3508. MR 0250269 (40:3508)
  • [7] V. V. Filippov, On the perfect image of a paracompact p-space, Dokl. Akad. Nauk SSSR 176 (1967), 533-535 = Soviet Math. Dokl. 8 (1967), 1151-1153. MR 36 #5903. MR 0222853 (36:5903)
  • [8] J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
  • [9] K. Morita, Products of normal spaces with metric spaces, Math. Ann. 154 (1964), 365-382. MR 29 #2773. MR 0165491 (29:2773)
  • [10] -, Some properties of M-spaces, Proc. Japan Acad. 43 (1967), 869-872. MR 37 #3517. MR 0227933 (37:3517)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54D20

Retrieve articles in all journals with MSC: 54D20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0346746-4
Keywords: Metric spaces, paracompact p-spaces, developable spaces, sequences of open covers, subparacompact spaces
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society