On spread and condensations
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- by A. V. Arhangelskii PDF
- Proc. Amer. Math. Soc. 124 (1996), 3519-3527 Request permission
Abstract:
A space $X$ has a property ${\mathcal {P}}$ strictly if every finite power of $X$ has ${\mathcal {P}}$. A condensation is a one-to-one continuous mapping onto. For Tychonoff spaces, the following results are established. If the strict spread of $X$ is countable, then $X$ can be condensed onto a strictly hereditarily separable space. If $s(C_{p}(X))\leq \omega$, then $C_{p}(X)$ can be condensed onto a strictly hereditarily separable space, and therefore, every compact subspace of $C_{p}(X)$ is strictly hereditarily separable. Under $(MA+\neg CH)$, if $G$ is a topological group such that $s(C_{p}(G))\leq \omega$, then $G$ is strictly hereditarily Lindelöf and strictly hereditarily separable.References
- A. V. Arkhangel′skiĭ, Topological function spaces, Mathematics and its Applications (Soviet Series), vol. 78, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian by R. A. M. Hoksbergen. MR 1144519, DOI 10.1007/978-94-011-2598-7
- A. V. Arkhangel′skiĭ, On hereditarily Lindelöf spaces of continuous functions, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3 (1989), 67–69 (Russian); English transl., Moscow Univ. Math. Bull. 44 (1989), no. 3, 67–69. MR 1029736
- A. V. Arhangel′skiĭ, The structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk 33 (1978), no. 6(204), 29–84, 272 (Russian). MR 526012
- A. V. Arhangel′skiĭ, Spaces that are elongated to the left, Vestnik Moskov. Univ. Ser. I Mat. Meh. 5 (1977), 30–36 (Russian, with English summary). MR 0482689
- Arhangel’skii A.V. and A. Bella, A few observations on topological spaces with small diagonal, Zbornik radova Filozofskog faculteta u Nis̆u Ser. Mat. 6 (1992), 211-213.
- Arhangel’skii A.V. and Fedorchuk V.V., On condensations of countably compact spaces onto compacta, Fund. Prikl. Mat. 1 (1995), 871–880. (Russian)
- A. V. Arkhangel′skiĭ and V. I. Ponomarev, Fundamentals of general topology, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1984. Problems and exercises; Translated from the Russian by V. K. Jain; With a foreword by P. Alexandroff [P. S. Aleksandrov]. MR 785749
- A. V. Arhangel′skii and V. V. Tkačuk, Calibers and point-finite cellularity of the space $Cp(X)$ and some questions of S. Gul′ko and M. Husek, Topology Appl. 23 (1986), no. 1, 65–73. MR 849094, DOI 10.1016/0166-8641(86)90017-9
- Asanov M., Cardinal invariants of spaces of continuous functions, Modern Topology and Set Theory, Izhevskij Universitet, Izhevsk, 1979, pp. (8-12). (Russian)
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- A. Hajnal and I. Juhász, On hereditarily $\alpha$-Lindelöf and $\alpha$-separable spaces. II, Fund. Math. 81 (1973/74), no. 2, 147–158. MR 336705, DOI 10.4064/fm-81-2-147-158
- A. Hajnal and I. Juhász, A separable normal topological group need not be Lindelöf, General Topology and Appl. 6 (1976), no. 2, 199–205. MR 431086, DOI 10.1016/0016-660X(76)90033-7
- M. Hušek, Topological spaces without $\kappa$-accessible diagonal, Comment. Math. Univ. Carolinae 18 (1977), no. 4, 777–788. MR 515009
- Kenneth Kunen, Strong $S$ and $L$ spaces under $MA$, Set-theoretic topology (Papers, Inst. Medicine and Math., Ohio Univ., Athens, Ohio, 1975–1976) Academic Press, New York, 1977, pp. 265–268. MR 0440487
- S. Negrepontis, Banach spaces and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 1045–1142. MR 776642
- Judy Roitman, The spread of regular spaces, General Topology and Appl. 8 (1978), no. 1, 85–91. MR 493957, DOI 10.1016/0016-660X(78)90020-X
- Judy Roitman, Basic $S$ and $L$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 295–326. MR 776626
- B. Šapirovskiĭ, Discrete subspaces of topological spaces. Weight, tightness and Suslin number, Dokl. Akad. Nauk SSSR 202 (1972), 779–782 (Russian). MR 0292012
- Stevo Todorčević, Forcing positive partition relations, Trans. Amer. Math. Soc. 280 (1983), no. 2, 703–720. MR 716846, DOI 10.1090/S0002-9947-1983-0716846-0
- Stevo Todorčević, Some applications of $S$ and $L$ combinatorics, The work of Mary Ellen Rudin (Madison, WI, 1991) Ann. New York Acad. Sci., vol. 705, New York Acad. Sci., New York, 1993, pp. 130–167. MR 1277886, DOI 10.1111/j.1749-6632.1993.tb12530.x
- N. V. Velichko, Weak topology of spaces of continuous functions, Mat. Zametki 30 (1981), no. 5, 703–712, 797 (Russian). MR 640070
- Phillip Zenor, Hereditary ${\mathfrak {m}}$-separability and the hereditary ${\mathfrak {m}}$-Lindelöf property in product spaces and function spaces, Fund. Math. 106 (1980), no. 3, 175–180. MR 584491, DOI 10.4064/fm-106-3-175-180
Additional Information
- A. V. Arhangelskii
- Affiliation: Chair of General Topology and Geometry, Mech.-Math. Faculty, Moscow University, Moscow 119899, Russia (June 15–December 31); Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701 (January 1–June 15)
- Email: aarhange@oucsace.cs.ohiou.edu
- Received by editor(s): April 7, 1995
- Additional Notes: The author was partially supported by NSF grant DMS-9312363.
- Communicated by: Franklin D. Tall
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 3519-3527
- MSC (1991): Primary 54A25, 54C35, 54A35
- DOI: https://doi.org/10.1090/S0002-9939-96-03605-2
- MathSciNet review: 1353369