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property $(a)$; dominating families; small cardinals; inner models of measurability
Generalizations of earlier negative results on Property $(a)$ are proved and two questions on an $(a)$-version of Jones' Lemma are posed. We discuss these questions in the realm of locally compact spaces. Using dominating families of functions as a tool, we prove that under the assumptions ``$2^\omega$ is regular'' and ``$2^\omega < 2^{\omega_1}$'' the existence of a $T_1$ separable locally compact $(a)$-space with an uncountable closed discrete subset implies the existence of inner models with measurable cardinals. We also use cardinal invariants such as $\frak d$ to prove results in the class of locally compact spaces that strengthen, in such class, the negative results mentioned above.
[C] Comfort W.W.: Cofinal families in certain function spaces. Comment. Math. Univ. Carolinae 29 4 (1988), 665-675. MR 0982784 | Zbl 0666.54002
[CS] Cummings J., Shelah S.: Cardinal invariants above the continuum. Ann. Pure Appl. Logic 75 3 (1995), 251-268. MR 1355135 | Zbl 0835.03013
[E] Engelking R.: General Topology. Heldermann Verlag, Sigma Series in Pure Mathematics 6, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[H] Hodel R.: Cardinal functions I. in: Handbook of Set-theoretic Topology, North-Holland, Amsterdam, 1984, pp.1-61. MR 0776620 | Zbl 0559.54003
[JMS] Just W., Matveev M.V., Szeptycki P.J.: Some results on property $(a)$. Topology Appl. 100 1 (2000), 103-111. MR 1731705 | Zbl 0944.54014
[JP] Jech T., Prikry K.: Cofinality of the partial ordering of functions from $ømega _1$ into $ømega $ under eventual domination. Math. Proc. Cambridge Philos. Soc. 95 1 (1984), 25-32. MR 0727077
[M94] Matveev M.V.: Absolutely countably compact spaces. Topology Appl. 58 1 (1994), 81-92. MR 1280711 | Zbl 0801.54021
[M97] Matveev M.V.: Some questions on property $(a)$. Questions Answers Gen. Topology 15 2 (1997), 103-111. MR 1472172 | Zbl 1002.54016
[SV] Szeptycki P.J., Vaughan J.E.: Almost disjoint families and property $(a)$. Fund. Math. 158 3 (1998), 229-240. MR 1663330 | Zbl 0933.54005
[vD] van Douwen E.K.: The integers and topology. in: Handbook of Set-theoretic Topology, North-Holland, Amsterdam, 1984, pp.111-167. MR 0776622 | Zbl 0561.54004
[vMR] van Mill J., Reed G.M. (editors): Open Problems in Topology. North-Holland, Amsterdam, 1990. MR 1078636
[W] Watson W.S.: Separation in countably paracompact spaces. Trans. Amer. Math. Soc. 290 2 (1985), 831-842. MR 0792831 | Zbl 0583.54013
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