Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
$G_\delta$-diagonal; property $(DC(\omega_1))$; cardinal; DCCC
$G_\delta$-diagonal; property $(DC(\omega_1))$; cardinal; DCCC
Summary:
We prove that if $X$ is a first countable space with property $(DC(\omega_1))$ and with a $G_\delta$-diagonal then the cardinality of $X$ is at most $\mathfrak c$. We also show that if $X$ is a first countable, DCCC, normal space then the extent of $X$ is at most $\mathfrak c$.
References:
[1] Aiken L.P.: Star-covering properties: generalized $\Psi$-spaces, countability conditions, reflection. Topology Appl. 158 (2011), no. 13, 1732–1737. DOI 10.1016/j.topol.2011.06.032 | MR 2812483 | Zbl 1223.54029
[2] Arhangel'skii A.V., Buzyakova R.Z.: The rank of the diagonal and submetrizability. Comment. Math. Univ. Carolin. 47 (2006), no. 4, 585–597. MR 2337413 | Zbl 1150.54335
[3] Arhangel'skii A.V., Burke D.K.: Spaces with a regular $G_\delta$-diagonal. Topology Appl. 153 (2006), no. 11, 1917–1929. DOI 10.1016/j.topol.2005.07.013 | MR 2227036 | Zbl 1117.54004
[4] Buzyakova R.Z.: Cardinalities of ccc-spaces with regular $G_\delta$-diagonals. Topology Appl. 153 (2006), no. 11, 1696–1698. DOI 10.1016/j.topol.2005.06.004 | MR 2227022
[5] Engelking R.: General Topology. Helderman, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[6] Ginsburg J., Woods R.G.: A cardinal inequality for topological spaces involving closed discrete sets. Proc. Amer. Math. Soc. 64 (1977), no. 2, 357–360. DOI 10.1090/S0002-9939-1977-0461407-7 | MR 0461407 | Zbl 0398.54002
[7] Ikenaga S.: Topological concept between Lindelöf and pseudo-Lindelöf. Research Reports of Nara National College of Technology 26 (1990), 103–108.
[8] Kunen K., Vaughan J.: Handbook of Set-theoretic Topology. North Holland, Amsterdam, 1984. MR 0776619 | Zbl 0674.54001
[9] Matveev M.: A survey on star covering properties. Topology Atlas, 1998.
[10] Porter J.R., Woods R.G.: Feebly compact spaces, Martin's axiom, and “diamond”. Topology Proc. 9 (1984), 105–121. MR 0781555 | Zbl 0565.54006
[11] Shakhmatov D.B.: No upper bound for cardinalities of Tychonoff c.c.c. spaces with a $G_\delta $-diagonal exists. Comment. Math. Univ. Carolin. 25 (1984), no. 4, 731–746. MR 0782022
[12] Uspenskij V.V.: A large $F_\sigma$-discrete Fréchet space having the Souslin property. Comment. Math. Univ. Carolin. 25 (1984), no. 2, 257–260. MR 0768812
[13] Xuan W.F., Shi W.X.: A note on spaces with a rank $3$-diagonal. Bull. Aust. Math. Soc. 90 (2014), no. 3, 521–524. DOI 10.1017/S0004972714000318 | MR 3270766 | Zbl 1305.54036
Search
Partner of
