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An irrational problem

100%
Tall F.
Fundamenta Mathematicae
|
2002
|
tom 175
|
nr 3
259-269
EN
Given a topological space ⟨X,𝓣⟩ ∈ M, an elementary submodel of set theory, we define $X_M$ to be X ∩ M with topology generated by ${U ∩ M: U ∈ 𝓣 ∩ M}$. 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 $X = X_M$.
2
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Lindelöf indestructibility, topological games and selection principles

64%
Scheepers M., Tall F.
Fundamenta Mathematicae
|
2010
|
tom 210
|
nr 1
1-46
EN
Arhangel'skii proved that if a first countable Hausdorff space is Lindelöf, then its cardinality is at most $2^{ℵ₀}$. 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, are essential tools in our investigations.
3
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Locally compact perfectly normal spaces may all be paracompact

64%
Larson P., Tall F.
Fundamenta Mathematicae
|
2010
|
tom 210
|
nr 3
285-300
EN
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 At a crucial step we use some still unpublished results announced by Todorcevic.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 Modulo these same unpublished results, we also show that if it is consistent that there is a supercompact cardinal, it is consistent that every locally compact space with a hereditarily normal square is metrizable. We also solve a problem raised by the second author, proving it consistent with ZFC that every first countable hereditarily normal countable chain condition space is hereditarily separable.
4
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Errata to "Lindelöf indestructibility, topological games and selection principles" (Fund. Math. 210 (2010), 1-46)

64%
Scheepers M., Tall F.
Fundamenta Mathematicae
|
2014
|
tom 224
|
nr 2
203-204
EN
In [Fund. Math. 210 (2010), 1-46] we claimed the truth of two statements, one now known to be false and a second lacking a proof. In this "Errata" we report these matters in the interest of setting the record straight on the status of these claims.
5
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More reflections on compactness

64%
Junqueira L., Tall F.
Fundamenta Mathematicae
|
2003
|
tom 176
|
nr 2
127-141
EN
We consider the question of when $X_M = X$, where $X_M$ is the elementary submodel topology on X ∩ M, especially in the case when $X_M$ is compact.
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