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)

 
 

 

Two classes of Fréchet-Urysohn spaces


Author: Alan Dow
Journal: Proc. Amer. Math. Soc. 108 (1990), 241-247
MSC: Primary 54E35; Secondary 03E35, 03E75, 54A35
DOI: https://doi.org/10.1090/S0002-9939-1990-0975638-7
MathSciNet review: 975638
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Arhangel'skii introduced five classes of spaces, $ {\alpha _i}$-spaces $ \left( {i < 5} \right)$, which are important in the study of products of Fréchet-Urysohn spaces. For each $ i < 5$, each $ {\alpha _i}$-space is an $ {\alpha _{i + 1}}$-space and it follows from the continuum hypothesis that there are countable $ {\alpha _{i + 1}}$-spaces which are not $ {\alpha _i}$-spaces. A $ v$-space ($ w$-space) is a Fréchet-Urysohn $ {\alpha _1}$-space ( $ {\alpha _2}$-space). We show that there is a model of set theory in which each $ {\alpha _2}$-space ($ w$-space) is an $ {\alpha _1}$-space ($ v$-space).


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

  • [A1] A. V. Arhangel'skii, The frequency spectrum of a topological space and the classification of spaces, Dokl. Akad. Nauk. SSSR 206 (1972), 265-268 = Sov. Math. Dokl. 13 (1972), 265-268. MR 0394575 (52:15376)
  • [A2] -, The frequency spectrum of a topological space and the product operation, Trudy Mosk. Mat. Obs. 40 (1979) = Transl. Moscow Math. Soc. (1981), Issue 2, 163-200.
  • [DS] A. Dow and J. Steprans.
  • [G] G. Gruenhage, Infinite games and generalizations of first countable spaces, Gen. Top. Appl. 6 (1976), 339-352. MR 0413049 (54:1170)
  • [L] R. Laver, On the consistency of Borel's conjecture, Acta Math. 137 (1976), 151-169. MR 0422027 (54:10019)
  • [LNy] R. Levy and P. Nyikos, Families in $ \beta \omega $ whose union is regular open, preprint.
  • [N] T. Nogura, A counterexample for a problem of Arhangel'skii concerning the product of Fréchet spaces, Top. Appl. 25 (1987), 75-80. MR 874979 (88e:54052)
  • [Ny1] P. Nyikos, $ {}^\omega \omega $ and the Fréchet-Urysohn property, in preparation.
  • [Ny2] -, The Cantor tree and the Fréchet-Urysohn property, preprint, 1987.
  • [O] R. C. Olson, Bi-quotient maps and countably bisequential spaces, Gen. Top. Appl. 4 (1974), 1-28. MR 0365463 (51:1715)
  • [S1] S. Shelah, Proper forcing, Springer Lecture Notes, 1982. MR 675955 (84h:03002)
  • [S2] -, Cardinal invariants of the continuum, Axiomatic Set Theory, Eds. J. Baumgartner, D. A. Martin, S. Shelah, Contemp. Math. Amer. Math. Soc., Providence, R.I., 1986.
  • [Sh] Sharma, Some characterizations of $ W$-spaces and $ w$-spaces, Gen. Top. Appl. 9 (1978) 289-293. MR 510910 (80a:54046)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54E35, 03E35, 03E75, 54A35

Retrieve articles in all journals with MSC: 54E35, 03E35, 03E75, 54A35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-0975638-7
Keywords: Fréchet-Urysohn space, $ v$-space, $ w$-space
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society