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An irrational problem

Franklin D. Tall — 2002

Fundamenta Mathematicae

Given a topological space ⟨X,⟩ ∈ M, an elementary submodel of set theory, we define X M to be X ∩ M with topology generated by U M : U M . Suppose X M is homeomorphic to the irrationals; must X = X M ? We have partial results. We also answer a question of Gruenhage by showing that if X M is homeomorphic to the “Long Cantor Set”, then X = X M .

Locally compact perfectly normal spaces may all be paracompact

Paul B. LarsonFranklin D. Tall — 2010

Fundamenta Mathematicae

We work towards establishing that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. At a crucial step we use some still unpublished results announced by Todorcevic. Modulo this and the large cardinal, this answers a question of S. Watson. Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space...

Lindelöf indestructibility, topological games and selection principles

Marion ScheepersFranklin D. Tall — 2010

Fundamenta Mathematicae

Arhangel’skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most 2 . Such a clean upper bound for Lindelöf spaces in the larger class of spaces whose points are G δ has been more elusive. In this paper we continue the agenda started by the second author, [Topology Appl. 63 (1995)], of considering the cardinality problem for spaces satisfying stronger versions of the Lindelöf property. Infinite games and selection principles, especially the Rothberger property,...

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