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On a Theorem of Arhangel'skiĭ Concerning Lindelöf P-Spaces

Published online by Cambridge University Press:  20 November 2018

R. E. Hodel
R. E. Hodel*
Affiliation:
Duke University, Durham, North Carolina
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1. Introduction. In [4] Arhangel'skiĭ proved the remarkable result that every regular space which is hereditarily a Lindelöf p-space has a countable base. As a consequence of the main theorem in this paper, we obtain an analogue of Arhangel'skiĭs result, namely that every regular space which is hereditarily an ℵi-compact strong ∑-space has a countable net. Under the assumption of the generalized continuum hypothesis (GCH), the main theorem also yields an affirmative answer to Problem 2 in Arhangel'skiĭs paper.

In § 3 we introduce and study a new cardinal function called the discreteness character of a space. The definition is based on a property first studied by Aquaro in [1], and for the class of T1 spaces it extends the concept of Kicompactness to higher cardinals.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 2 , 01 April 1975 , pp. 459 - 468
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Aquaro, G., Point-countable open coverings in countably compact spaces, General Topology and Its Relations to Modern Analysis and Algebra II (Academia, Prague, 1966), 39–41.Google Scholar
2. Arhangefskiï, A. V., On a class of spaces containing all metric and all locally bicompact spaces, Amer. Math. Soc. Transi. 2 (1970), 1–39.Google Scholar
3. Arhangefskiï, A. V., An addition theorem for the weight of spaces lying in bicompacta, Dokl. Akad. Nauk SSSR 126 (1959), 239–241.Google Scholar
4. Arhangefskiï, A. V., On hereditary properties, General Topology and Appl. 3 (1973), 39–46.Google Scholar
5. Burke, D. K., On p-spaces and wA-spaces, Pacific J. Math. 35 (1970), 285–296.Google Scholar
6. Comfort, W. W., A survey of cardinal invariants, General Topology and Appl. 1 (1971), 163–200.Google Scholar
7. Filippov, V. V., On feathered paracompacta, Soviet Math. Dokl. 9 (1968), 161–164.Google Scholar
8. Hajnal, A. and Juhâsz, I., Discrete subspaces of topological spaces, Indag. Math. 29 (1967), 343–356.Google Scholar
9. Hodel, R. E., On the weight of a topological space, Proc. Amer. Math. Soc. 43 (1974), 470–474.Google Scholar
10. Hodel, R. E., Extensions of metrization theorems to higher cardinality (to appear in Fund. Math.).Google Scholar
11. Holsztynski, V., Hausdorff spaces of minimal weight, Soviet Math. Dokl. 7 (1966), 667–668.Google Scholar
12. Juhâsz, I., Cardinal functions in topology (Mathematical Centre, Amsterdam, 1971).Google Scholar
13. Kuratowski, K., Topology, Vol. 1 (Academic Press, New York, 1966).Google Scholar
14. Michael, E., A note on paracompact spaces, Proc. Amer. Math. Soc. 4 (1953), 831–838.Google Scholar
15. Michael, E., On Nagamis C-spaces and some related matters, Proceedings of the Washington State University Conference on General Topology (1970), 13–19.Google Scholar
16. Miscenko, A. S., Spaces with point-countable bases, Soviet Math. Dokl. 3 (1962), 855–858.Google Scholar
17. Nagami, K., C-spaces, Fund. Math. 65 (1969), 169–192.Google Scholar
18. Nagata, J., A note on Filippov1 s theorem, Proc. Japan Acad. 45 (1969), 30–33.Google Scholar
19. Ponomarev, V. I., Metrizability of a finally compact p-space with a point-countable base, Soviet Math. Dokl. 8 (1967), 765–768.Google Scholar
20. Stephenson, R. M., Jr., Discrete subsets of perfectly normal spaces, Proc. Amer. Math. Soc. 34 (1972), 605–608.Google Scholar