Bounds on the extent of a topological space
Abstract
The extent $e(X)$ of a topological space $X$ is the supremum of sizes of closed discrete subspaces of $X$. Assuming that $X$ belongs to some class of topological spaces, we bound $e(X)$ by
other cardinal characteristics of $X$, for instance Lindel\"of number, spread or density.
References
A.V. Arhangel’skii, D-spaces and covering properties, Topology Appl., 146–147 (2005), 437–449.
O.T. Alas, L.R. Junqueira, J. van Mill, V. Tkachuk, R.G. Wilson, On the extent of star countable spaces, Cent. Eur. J. Math., 9 (2011), №3, 603–615, staff.science.uva.nl/j.vanmill/papers/papers2011/al-ju-vmtk-wi.pdf
T. Banakh, A. Ravsky, Verbal covering properties of topological spaces, Topology Appl., 201 (2016), 181–205.
M. Bonanzinga, M. Matveev, Problems on star-covering properties, in Open problems in Topology II (ed.: E. Pearl), Elsevier, 2007.
D. Burke, Covering properties, in K. Kunen, J.E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier, 1984, 347–422.
P. de Caux, A collectionwise normal weakly θ-refinable Dowker space which is neither irreducible nor realcompact, Topology Proceedings, I (1976), 67–77.
E.K. van Douwen, W. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math., 81 (1979), 371–377.
E.K. van Douwen, H.H. Wicke, A real, weird topology on the reals, Houston J. Math., 3 (1977), №1, 141–152.
R. Engelking, General topology, 2nd ed., Heldermann, Berlin, 1989.
G. Gruenhage, Generalized metric spaces, in K. Kunen, J. E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier, 1984, 423–501.
G. Gruenhage, A survey of D-spaces, Set theory and its applications, 13–28, Contemp. Math., 533, Amer. Math. Soc., Providence, RI, 2011, auburn.edu/ gruengf/papers/dsurv7.pdf
G.R. Hiremath, On star with Lindel¨of center property, J. Indian Math. Soc., 59 (1993), 227–242.
F.B. Jones, Concerning normal and completely normal spaces, Bull. Amer. Math. Soc., 43 (1937), 671–677.
R. Levy, M. Matveev, Weak extent in normal spaces, Comment. Math. Univ. Carolin., 46 (2005), №3, 497–501.
D.J. Lutzer, Another property of the Sorgenfrey line, Compositio Mathematica, 24 (1972), №3, 359–363, eudml.org/doc/89127
Dan Ma, Weakly Lindel¨of spaces, dantopology.wordpress.com/2014/02/28/weakly-lindelof-spaces
J.D. Mashburn, A note on irreducibility and weak covering properties, Topology Proc., 9 (1984), №2, 339–352, topology.auburn.edu/tp/reprints/v09/tp09212.pdf
M.V. Matveev, How weak is weak extent? Topology Appl., 119 (2002), №2, 229–232, arxiv.org/abs/math/0006198
M.V. Matveev, A survey on star covering properties, Topology Atlas, preprint 330, 1998, at.yorku.ca/v/a/a/a/19.htm
C.J.G. Morgan, S.G. da Silva, Constraining extent by density: on generalizations of normality and countable paracompactness, Bol. Soc. Mat. Mexicana, 18, (2012), №3.
S. Mr´owka, On completely regular spaces, Fund. Math., 41 (1954), 105–106.
J.E. Vaughan, Countably compact and sequentially compact spaces, in K. Kunen, J.E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier, 1984, 569–602.
Z. Yu, Z. Yun, D-spaces, aD-spaces, and finite unions, Topology Appl., 156 (2009), 1459–1462.
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