Countable size nonmetrizable spaces which are stratifiable and $\kappa$-metrizable
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Abstract:
K. Tamano asked the following question. Is a stratifiable space metrizable if it is $\kappa$-metrizable? To answer this question, we show that the class of stratifiable $\kappa$-metrizable spaces is much wider than that of metrizable spaces. In fact, we describe two very distinct classes of countable, stratifiable $\kappa$-metrizable spaces which are not metrizable. One of them has no nontrivial convergent sequence. The other is bisequential but not a w-space. In addition, we give a characterization of $\kappa$-metrizability of countable spaces defined by the Cantor tree and note a topological property of monotonically normal $\kappa$-metrizable spaces.References
- A. V. Arkhangel′skiĭ, Hurewicz spaces, analytic sets and fan tightness of function spaces, Dokl. Akad. Nauk SSSR 287 (1986), no. 3, 525–528 (Russian). MR 837289 A. N. Dranishnikov, Simultaneous annihilation of families of closed sets, $\kappa$-metrizable and stratifiable spaces, Soviet Math. Dokl. 19 (1978), 1466-1469.
- Ryszard Engelking, General topology, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR 1039321
- J. Gerlits and Zs. Nagy, Some properties of $C(X)$. I, Topology Appl. 14 (1982), no. 2, 151–161. MR 667661, DOI 10.1016/0166-8641(82)90065-7
- Gary Gruenhage, Generalized metric spaces, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 423–501. MR 776629
- R. W. Heath, An easier proof that a certain countable space is not stratifiable. , Proc. Washington State Univ. Conf. on General Topology (Pullman, Wash., 1970) Washington State University, Department of Mathematics, Pi Mu Epsilon, Pullman, Wash., 1970, pp. 56–59. MR 0266135
- R. W. Heath, D. J. Lutzer, and P. L. Zenor, Monotonically normal spaces, Trans. Amer. Math. Soc. 178 (1973), 481–493. MR 372826, DOI 10.1090/S0002-9947-1973-0372826-2
- E. A. Michael, A quintuple quotient quest, General Topology and Appl. 2 (1972), 91–138. MR 309045, DOI 10.1016/0016-660X(72)90040-2
- Peter J. Nyikos, Classes of compact sequential spaces, Set theory and its applications (Toronto, ON, 1987) Lecture Notes in Math., vol. 1401, Springer, Berlin, 1989, pp. 135–159. MR 1031771, DOI 10.1007/BFb0097337
- Peter Nyikos, The Cantor tree and the Fréchet-Urysohn property, Papers on general topology and related category theory and topological algebra (New York, 1985/1987) Ann. New York Acad. Sci., vol. 552, New York Acad. Sci., New York, 1989, pp. 109–123. MR 1020779, DOI 10.1111/j.1749-6632.1989.tb22391.x
- P. J. Nyikos and S. Purisch, Monotone normality and paracompactness in scattered spaces, Papers on general topology and related category theory and topological algebra (New York, 1985/1987) Ann. New York Acad. Sci., vol. 552, New York Acad. Sci., New York, 1989, pp. 124–137. MR 1020780, DOI 10.1111/j.1749-6632.1989.tb22392.x
- Haruto Ohta and Ken-ichi Tamano, Perfect images of zero-dimensional $\sigma$-spaces, Kobe J. Math. 7 (1990), no. 1, 89–108. MR 1077573
- E. G. Pytkeev, Sequentiality of spaces of continuous functions, Uspekhi Mat. Nauk 37 (1982), no. 5(227), 197–198 (Russian). MR 676634
- Masami Sakai, Property C′′and function spaces, Proc. Amer. Math. Soc. 104 (1988), no. 3, 917–919. MR 964873, DOI 10.1090/S0002-9939-1988-0964873-0 E. V. Ščepin, On $\kappa$-metrizable spaces, Math. USSR-Izv. 14 (1980), 407-440.
- P. L. Sharma, Some characterizations of $W$-spaces and $w$-spaces, General Topology Appl. 9 (1978), no. 3, 289–293. MR 510910, DOI 10.1016/0016-660x(78)90032-6
- J. Suzuki, K. Tamano, and Y. Tanaka, $\kappa$-metrizable spaces, stratifiable spaces and metrization, Proc. Amer. Math. Soc. 105 (1989), no. 2, 500–509. MR 933521, DOI 10.1090/S0002-9939-1989-0933521-9
- Ken-ichi Tamano, Closed images of metric spaces and metrization, Proceedings of the 1985 topology conference (Tallahassee, Fla., 1985), 1985, pp. 177–186. MR 851211
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 265-273
- MSC: Primary 54D15; Secondary 54E20, 54E35
- DOI: https://doi.org/10.1090/S0002-9939-1994-1231043-3
- MathSciNet review: 1231043