Abstract
In this note, we show that if for any transitive neighborhood assignment φ for X there is a point-countable refinement ℱ such that for any non-closed subset A of X there is some V ∈ ℱ such that |V ∩ A| ⩾ ω, then X is transitively D. As a corollary, if X is a sequential space and has a point-countable wcs*-network then X is transitively D, and hence if X is a Hausdorff k-space and has a point-countable k-network, then X is transitively D. We prove that if X is a countably compact sequential space and has a pointcountable wcs*-network, then X is compact. We point out that every discretely Lindelöf space is transitively D. Let (X, τ) be a space and let (X, ℐ) be a butterfly space over (X, τ). If (X, τ) is Fréchet and has a point-countable wcs*-network (or is a hereditarily meta-Lindelöf space), then (X, ℐ) is a transitively D-space.
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References
A. V. Arhangel’skii: D-spaces and covering properties. Topology Appl. 146–147 (2005), 437–449.
A.V. Arhangel’skii: D-spaces and finite unions. Proc. Am. Math. Soc. 132 (2004), 2163–2170.
A. V. Arhangel’skii: A generic theorem in the theory of cardinal invariants of topological spaces. Commentat. Math. Univ. Carol. 36 (1995), 303–325.
A. V. Arhangel’skii, R. Z. Buzyakova: Addition theorems and D-spaces. Commentat. Math. Univ. Carol. 43 (2002), 653–663.
A. V. Arhangel’skii, R. Z. Buzyakova: On linearly Lindelöf and strongly discretely Lindelöf spaces. Proc. Am. Math. Soc. 127 (1999), 2449–2458.
C. R. Borges, A. C. Wehrly: A study of D-spaces. Topol. Proc. 16 (1991), 7–15.
D. K. Burke: Weak-bases and D-spaces. Commentat. Math. Univ. Carol. 48 (2007), 281–289.
R. Z. Buzyakova: On D-property of strong Σ-spaces. Commentat. Math. Univ. Carol. 43 (2002), 493–495.
E. K. van Douwen, W. F. Pfeffer: Some properties of the Sorgenfrey line and related spaces. Pac. J. Math. 81 (1979), 371–377.
R. Engelking: General Topology. Sigma Series in Pure Mathematics, Vol. 6, revised ed. Heldermann, Berlin, 1989.
G. Gruenhage: A note on D-spaces. Topology Appl. 153 (2006), 2229–2240.
G. Gruenhage: Generalized Metric Spaces. In: Handbook of Set-Theoretic Topology (K. Kunen, J. E. Vaughan, eds.). North-Holland, Amsterdam, 1984, pp. 423–501.
G. Gruenhage: Submeta-Lindelöf implies transitively D. Preprint.
G. Gruenhage, E. Michael, Y. Tanaka: Spaces determined by point-countable covers. Pac. J. Math. 113 (1984), 303–332.
G. Gruenhage: A survey of D-spaces. Contemporary Mathematics 533 (L. Babinkostova, ed.). American Mathematical Society, Providence, 2011, pp. 13–28.
H.F. Guo, H. Junnila: On spaces which are linearly D. Topology Appl. 157 (2010), 102–107.
S. Lin: A note on D-spaces. Commentat. Math. Univ. Carol. 47 (2006), 313–316.
S. Lin: Point-Countable Covers and Sequence-Covering Mappings. Chinese Science Press, Beijing, 2002. (In Chinese.)
S. Lin, Y. Tanaka: Point-countable k-networks, closed maps and related results. Topology Appl. 59 (1994), 79–86.
E. Pearl: Linearly Lindelöf problems. In: Open Problems in Topology II (E. Pearl, ed.). Elsevier, Amsterdam, 2007, pp. 225–231.
L.-X. Peng: On spaces which are D, linearly D and transitively D. Topology Appl. 157 (2010), 378–384.
L.-X. Peng: The D-property of some Lindelöf spaces and related conclusions. Topology Appl. 154 (2007), 469–475.
L.-X. Peng: On linear neighborhood assignments and dually discrete spaces. Topology Appl. 155 (2008), 1867–1874.
L.-X. Peng: A special point-countable family that makes a space to be a D-space. Adv. Math. (China) 37 (2008), 724–728.
L.-X. Peng: A note on D-spaces and infinite unions. Topology Appl. 154 (2007), 2223–2227.
L.-X. Peng: On weakly monotonically monolithic spaces. Commentat. Math. Univ. Carol. 51 (2010), 133–142.
L.-X. Peng: A note on transitively D and D-spaces. Houston J. Math. To appear.
L.-X. Peng, Franklin D. Tall: A note on linearly Lindelöf spaces and dual properties. Topol. Proc. 32 (2008), 227–237.
F. Siwiec: On defining a space by a weak base. Pac. J. Math. 52 (1974), 233–245.
L.A. Steen, Jr. J. A. Seebach: Counterexamples in Topology. 2nd edition. Springer, New York-Heidelberg-Berlin, 1978.
V.V. Tkachuk: Monolithic spaces and D-spaces revisited. Topology Appl. 156 (2009), 840–846.
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Research supported by Beijing Natural Science Foundation (Grant No. 1102002), supported by the National Natural Science Foundation of China (Grant No. 10971185), and sponsored by SRF for ROCS, SEM.
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Peng, LX. A note on transitively D-spaces. Czech Math J 61, 1049–1061 (2011). https://doi.org/10.1007/s10587-011-0047-5
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DOI: https://doi.org/10.1007/s10587-011-0047-5