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Keywords:
Baire spaces; resolvable spaces; almost resolvable spaces; almost-$\omega$-resolvable spaces; tightness; $\pi$-weight
Baire spaces; resolvable spaces; almost resolvable spaces; almost-$\omega$-resolvable spaces; tightness; $\pi$-weight
Summary:
We continue the study of almost-$\omega$-resolvable spaces beginning in A. Tamariz-Mascar'ua, H. Villegas-Rodr'{\i}guez, {\it Spaces of continuous functions, box products and almost-$\omega$-resoluble spaces\/}, Comment. Math. Univ. Carolin. {\bf 43} (2002), no. 4, 687--705. We prove in ZFC: (1) every crowded $T_0$ space with countable tightness and every $T_1$ space with $\pi$-weight $\leq \aleph _1$ is hereditarily almost-$\omega$-resolvable, (2) every crowded paracompact $T_2$ space which is the closed preimage of a crowded Fréchet $T_2$ space in such a way that the crowded part of each fiber is $\omega$-resolvable, has this property too, and (3) every Baire dense-hereditarily almost-$\omega$-resolvable space is $\omega$-resolvable. Moreover, by using the concept of almost-$\omega$-resolvability, we obtain two results due the first one to O. Pavlov and the other to V.I. Malykhin: (1) $V = L$ implies that every crowded Baire space is $\omega$-resolvable, and (2) $V = L$ implies that the product of two crowded spaces is resolvable. Finally, we prove that the product of two almost resolvable spaces is resolvable.
References:
[A] Alas O.T., Sanchis M., Tkachenko M.G., Tkachuk V.V., Wilson R.G.: Irresolvable and submaximal spaces: Homogeneity versus $\sigma$-discreteness and new ZFC examples. Topology Appl. 107 (2000), 259-273. MR 1779814 | Zbl 0984.54002
[Ar] Arkhangel'skii A.V.: Topological Function Spaces. Kluwer Academic Publishers, Dordrecht (1992). MR 1144519
[BM] Bella A., Malykhin V.I.: Tightness and resolvability. Comment. Math. Univ. Carolin. 39 (1998), 177-184. MR 1623014 | Zbl 0936.54004
[B] Bolstein R.: Sets of points of discontinuity. Proc. Amer. Math. Soc. 38 (1973), 193-197. DOI 10.1090/S0002-9939-1973-0312457-9 | MR 0312457 | Zbl 0232.54014
[CGF] Comfort W.W., García-Ferreira S.: Resolvability: a selective survey and some new results. Topology Appl. 74 (1996), 149-167. DOI 10.1016/S0166-8641(96)00052-1 | MR 1425934
[CF] Comfort W.W., Feng L.: The union of resolvable spaces is resolvable. Math. Japon. 38 (1993), 413-114. MR 1221007 | Zbl 0769.54002
[vD] van Douwen E.K.: Applications of maximal topologies. Topology Appl. 51 (1993), 125-139. DOI 10.1016/0166-8641(93)90145-4 | MR 1229708 | Zbl 0845.54028
[E1] El'kin A.G.: Decomposition of spaces. Soviet Math. Dokl. 10 (1969), 521-525. Zbl 0202.53701
[E2] El'kin A.G.: On the maximal resolvability of products of topological spaces. Soviet Math. Dokl. 10 (1969), 659-662. MR 0248726 | Zbl 0199.57302
[E3] El'kin A.G.: Resolvable spaces which are not maximally resolvable. Moscow Univ. Math. Bull. 24 (1969), 116-118. MR 0256331 | Zbl 0183.51204
[FL] Foran J., Liebnits P.: A characterization of almost resolvable spaces. Rend. Circ. Mat. di Palermo, Serie II XL (1991), 136-141. MR 1119751
[FM] Feng L., Masaveu O.: Exactly $n$-resolvable spaces and $ømega$-resolvability. Math. Japon. 50 (1999), 333-339. MR 1727655 | Zbl 0998.54026
[Gr] Gruenhage G.: Generalized metric spaces. Handbook of Set Theoretic-Topology, K. Kunen and J. Vaughan, Eds., North Holland, Amsterdam, New York, Oxford, Tokio, 1984. MR 0776629 | Zbl 0794.54034
[H] Hewitt E.: A problem of set-theoretic topology. Duke Math. J. 10 (1943), 306-333. DOI 10.1215/S0012-7094-43-01029-4 | MR 0008692 | Zbl 0060.39407
[Ho] Hodel R.: Cardinal functions I. 1-61 Handbook of Set-Theoretic Topology North-Holland (1984), Amsterdam-New-York-Oxford. MR 0776620 | Zbl 0559.54003
[I] Illanes A.: Finite and $ømega$-resolvability. Proc. Amer. Math. Soc. 124 (1996), 1243-1246. DOI 10.1090/S0002-9939-96-03348-5 | MR 1327020 | Zbl 0856.54010
[K] Katětov M.: On topological spaces containing no disjoint dense sets. Mat. Sbornik 21 (1947), 3-12. MR 0021679
[KST] Kunen K., Szymansky A., Tall F.: Baire irresolvable spaces and ideal theory. Annal Math. Silesiana 2 (14) (1986), 98-107. MR 0861505
[M1] Malykhin V.I.: On the resolvability of the product of two spaces and a problem of Katětov. Dokl. Akad. Nauk SSSR 222 (1975), 765-729. Zbl 0325.54017
[M2] Malykhin V.I.: Product of ultrafilters and irresolvable spaces. Mat. Sbornik 90 (132) (1973), 105-115. DOI 10.1070/SM1973v019n01ABEH001738
[Pa] Pavlov O.: On resolvability of topological spaces. Topology Appl. 126 (2002), 37-47. DOI 10.1016/S0166-8641(02)00004-4 | MR 1934251 | Zbl 1012.54004
[Pa1] Pavlov O.: Problems on resolvability. in Open Problems in Topology, II (Elsevier Publishers, 2007).
[P] Pytkeev E.G.: On maximally resolvable spaces. Proc. Steklov Inst. Math. 154 (1984), 225-230. Zbl 0557.54002
[TV] Tamariz-Mascarúa A., Villegas-Rodríguez H.: Spaces of continuous functions, box products and almost-$ømega$-resoluble spaces. Comment. Math. Univ. Carolin. 43 4 (2002), 687-705. MR 2045790
[Vi1] Villegas L.M.: On resolvable spaces and groups. Comment. Math. Univ. Carolin. 36 (1995), 579-584. MR 1364498 | Zbl 0837.22001
[Vi2] Villegas L.M.: Maximal resolvability of some topological spaces. Bol. Soc. Mat. Mexicana 5 (1999), 123-136. MR 1692526 | Zbl 0963.22001
[W] Willard S.: General Topology. Addison-Wesley Publishing Co. (1970), Reading, Mass.-London-Don Mills. MR 0264581 | Zbl 0205.26601
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