Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A symmetrizable space that is not perfect


Author: Dennis A. Bonnett
Journal: Proc. Amer. Math. Soc. 34 (1972), 560-564
MSC: Primary 54A25
DOI: https://doi.org/10.1090/S0002-9939-1972-0295275-9
MathSciNet review: 0295275
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An example is given of a Hausdorff symmetrizable space which has a closed subset that is not a $ {G_\delta }$-subset (thus, it is not perfect) and which is not subparacompact. This example is then used in the construction of a symmetrizable $ {T_1}$-space Y having a point $ {x_0}$ such that $ \{ {x_0}\} $ is not a $ {G_\delta }$-subset of Y.


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

  • [1] A. V. Arhangel'skiĭ, Behavior of metrizability under factor mapping, Dokl. Akad. Nauk SSSR 164 (1965), 247-250=Soviet Math. Dokl. 6 (1965), 1187-1190. MR 33 #697. MR 0192472 (33:697)
  • [2] -, 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)
  • [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] D. K. Burke, Subparacompact spaces, Proc. Washington State Univ. Conf. on General Topology (Pullman, Wash., 1970), Pi Mu Epsilon, Dept. of Math., Washington State University, Pullman, Wash., 1970, pp. 39-49. MR 42 #1066. MR 0266158 (42:1066)
  • [5] D. K. Burke and R. A. Stoltenberg, Some properties of $ \pi $-images of metric spaces (to appear).
  • [6] G. Hamel, Eine Basis aller Zahlen und die unstetigen Losungen der funktional gleichung: $ f(x + y) = f(x) + f(y)$, Math. Ann. 60 (1905), 459-462. MR 1511317
  • [7] R. W. Heath, Semi-metric and related spaces, Topology Conference, Arizona State University, Tempe, Ariz., 1967, pp. 153-161.
  • [8] S. I. Nedev, Symmetrizable spaces and final compactness, Dokl. Akad. Nauk SSSR 175 (1967), 532-534=Soviet Math. Dokl. 8 (1967), 890-892. MR 35 #7293. MR 0216460 (35:7293)
  • [9] J. Novák, Induktion partiell stetiger Funktionen, Math. Ann. 118 (1942), 449-461. MR 6, 164. MR 0011433 (6:164c)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54A25

Retrieve articles in all journals with MSC: 54A25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0295275-9
Keywords: Abstract spaces, symmetrizable, semimetrizable, $ {G_\delta }$-subset, subparacompact, perfect space
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society