Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-24T08:06:40.265Z Has data issue: false hasContentIssue false
  • English
  • Français

Productively Lindelöf Spaces May All Be D

Published online by Cambridge University Press:  20 November 2018

Franklin D. Tall
Franklin D. Tall*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4 e-mail: f.tall@utoronto.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We give easy proofs that (a) the Continuum Hypothesis implies that if the product of $X$ with every Lindelöf space is Lindelöf, then $X$ is a $D$-space, and (b) Borel's Conjecture implies every Rothberger space is Hurewicz.

Keywords

54D2054B1054D5554A2054A20productively LindelöfD-space, projectively σ-compactMengerHurewicz
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 1 , 01 March 2013 , pp. 203 - 212
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Alas, O. T., Aurichi, L. F., Junqueira, L. R., and Tall, F. D., Non-productively Lindelöf spaces and small cardinals. Houston J. Math., to appear.Google Scholar
[2] Alster, K., On spaces whose product with every Lindelöf space is Lindelöf. Colloq. Math. 54 (1987), no. 2, 171–178.Google Scholar
[3] Alster, K., On the class of all spaces of weight not greater than!1 whose Cartesian product with every Lindelöf space is Lindelöf. Fund. Math. 129 (1988), no. 2, 133–140.Google Scholar
[4] Arhangel'skiĭ, A. V., Projective -compactness,!1-caliber, and Cp-spaces. Topology Appl. 104 (2000), no. 1-3, 13–26. http://dx.doi.org/10.1016/S0166-8641(99)00011-5 Google Scholar
[5] Aurichi, L. F., D-spaces, topological games and selection principles. Topology Proc. 36 (2010), 107–122.Google Scholar
[6] Aurichi, L. F. and Tall, F. D., Lindelöf spaces which are indestructible, productive, or D. Topology Appl., to appear.Google Scholar
[7] Banakh, T. and Zdomskyy, L., Selection principles and infinite games on multicovered spaces. In: Selection principles and covering properties in toppology, Quad. Mat., 18, Dept. Math., Seconda Univ. Napoli, Caserta, 2006, pp. 1–51.Google Scholar
[8] Barr, M., Kennison, J. F., and Raphael, R., On productively Lindelöf spaces. Sci. Math. Jpn. 65 (2007), no. 3, 319–332.Google Scholar
[9] Blass, A., Combinatorial cardinal characteristics of the continuum. In: Handbook of Set Theory. Springer, 2010, pp. 395–489.Google Scholar
[10] Bonanzinga, M., Cammaroto, F., and Matveev, M., Projective versions of selection principles. Topology Appl. 157 (2010), no. 5, 874–893. http://dx.doi.org/10.1016/j.topol.2009.12.004 Google Scholar
[11] Borel, E., E. Sur la classification des ensembles de mesure nulle. Bull. Soc. Math. France 47 (1919), 97–125.Google Scholar
[12] van Douwen, E. K. and Pfeffer, W., Some properties of the Sorgenfrey line and related spaces. Pacific J. Math. 81 (1979), no. 2, 371–377.Google Scholar
[13] Eisworth, T., On D-spaces. In: Open Problems in Topology. II. Elsevier, Amsterdam, 2007, pp. 129–134.Google Scholar
[14] Engelking, R., General Topology. Second edition. Sigma Series in Pure Mathematics 6. Heldermann Verlag, Berlin, 1989.Google Scholar
[15] Gruenhage, G., A survey of D-spaces. In: Set Theory and its Applications, Contemp. Math., 533, American Mathematical Society, Providence, RI, 2011, pp. 13–28.Google Scholar
[16] Hansell, R. W., Descriptive topology. In: Recent Progress in General Topology. North-Holland, Amsterdam, 1992, pp. 275–315.Google Scholar
[17] Hurewicz, W., Uber eine Verallgemeinerung des Borelschen Theorems. Math. Zeit. 24 (1926), 401–421. http://dx.doi.org/10.1007/BF01216792 Google Scholar
[18] Just, W., Miller, A. W., M. Scheepers, and Szeptycki, P. J., The combinatorics of open covers. II. Topology Appl. 73 (1996), no. 3, 241–266. http://dx.doi.org/10.1016/S0166-8641(96)00075-2 Google Scholar
[19] Kočinac, L. D., Selection principles and continuous images. Cubo 8 (2000), no. 2, 23–31.Google Scholar
[20] Laver, R., On the consistency of Borel's conjecture. Acta. Math. 137 (1976), no. 3-4, 151–169. http://dx.doi.org/10.1007/BF02392416 Google Scholar
[21] Lawrence, L. B., The influence of a small cardinal on the product of a Lindelöf space and the irrationals. Proc. Amer. Math. Soc. 110 (1990), no. 2, 535–542.Google Scholar
[22] Michael, E. A., Paracompactness and the Lindelöf property in finite and countable Cartesian products. Compositio Math. 23 (1971), 199–214.Google Scholar
[23] Miller, A. W., Special subsets of the real line. In: Handbook of Set-Theoretic Topology. North-Holland, Amsterdam, 1984, pp. 201–233 Google Scholar
[24] Moore, J. T., Some of the combinatorics related to Michael's problem. Proc. Amer. Math. Soc. 127 (1999), no. 8, 2459–2467. http://dx.doi.org/10.1090/S0002-9939-99-04808-X Google Scholar
[25] Moore, J. T., A solution to the L-space problem. J. Amer. Math. Soc. 19 (2006), no. 3, 717–736. http://dx.doi.org/10.1090/S0894-0347-05-00517-5 Google Scholar
[26] Scheepers, M., A direct proof of a theorem of Telg´arsky. Proc. Amer. Math. Soc. 123 (1995), no. 11, 3483–3485.Google Scholar
[27] Scheepers, M., Topological games. In: Encyclopedia of General Topology. Elsevier, Amsterdam, 2004, pp. 439–442.Google Scholar
[28] Scheepers, M. and Tall, F. D., Lindelöf indestructibility, topological games and selection principles. Fund. Math. 210 (2010), no. 1, 1–46. http://dx.doi.org/10.4064/fm210-1-1 Google Scholar
[29] Tall, F. D., Lindelöf spaces which are Menger, Hurewicz, Alster, productive or D. Topology Appl. To appear.Google Scholar
[30] Tall, F. D. and Tsaban, B., A note on productively Lindelöf spaces. Topology Appl. 158 (2011), no. 11, 1239–1248. http://dx.doi.org/10.1016/j.topol.2011.04.003 Google Scholar
[31] Telg´arsky, R., On games of Topsøe. Math. Scand. 54 (1984), no. 1, 170–176.Google Scholar
[32] Tsaban, B. and Zdomskyy, L., Scales, fields, and a problem of Hurewicz. J. Eur. Math. Soc. 10 (2008), no. 3, 837–866. http://dx.doi.org/10.4171/JEMS/132 Google Scholar
[33] Tsaban, B. and Zdomskyy, L., Arhangel’ski sheaf amalgamations in topological groups. http://www.logic.univie.ac.at/_lzdomsky Google Scholar
[34] Zdomskyy, L., A semifilter approach to selection principles. Comment. Math. Univ. Carolin. 46 (2005), no. 3, 525–539.Google Scholar