Perfect Mappings and Spaces of Countable Type
Canadian journal of mathematics, Tome 22 (1970) no. 6, pp. 1208-1210
Voir la notice de l'article provenant de la source Cambridge University Press
In [1, p. 41, Theorem 3.10] Arhangel'skiï proved that the perfect image of a completely regular space of countable type is of countable type, and he asked [1, p. 60, problem 4] if a similar result held for regular or Hausdorff spaces. In this paper, it is proved that the perfect image of a space of countable type is of countable type, provided that the image is Hausdorff or regular. An affirmative answer to both of Arhangel'skiï's questions follows immediately from this. Arhangel'skiï made use of the Stone-Čech compactification in the proof of his result, but the proofs below are of a different nature.Let X be a topological space and let K ⊂ X. A collection of open sets is called a base at K provided that for every open set W ⊃ K there exists such that K ⊂ U ⊂ W. Clearly, we may assume that every member of contains K.
Vaughan, J. E. Perfect Mappings and Spaces of Countable Type. Canadian journal of mathematics, Tome 22 (1970) no. 6, pp. 1208-1210. doi: 10.4153/CJM-1970-138-3
@article{10_4153_CJM_1970_138_3,
author = {Vaughan, J. E.},
title = {Perfect {Mappings} and {Spaces} of {Countable} {Type}},
journal = {Canadian journal of mathematics},
pages = {1208--1210},
year = {1970},
volume = {22},
number = {6},
doi = {10.4153/CJM-1970-138-3},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1970-138-3/}
}
[1] 1. Arhangel'skiï, A. V., Bicompact sets and the topology of spaces, Trans. Moscow Math. Soc. 13 (1965), 1–62 ( = Trudy Moskov. Mat. Obsc. 13 (1965), 3-55). Google Scholar
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