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1
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Partitions of compact Hausdorff spaces

100%
Gruenhage G.
Fundamenta Mathematicae
|
1993
|
tom 142
|
nr 1
89-100
EN
Under the assumption that the real line cannot be covered by $ω_1$-many nowhere dense sets, it is shown that (a) no Čech-complete space can be partitioned into $ω_1$-many closed nowhere dense sets; (b) no Hausdorff continuum can be partitioned into $ω_1$-many closed sets; and (c) no compact Hausdorff space can be partitioned into $ω_1$-many closed $G_δ$-sets.
2
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The Arkhangel’skiĭ–Tall problem: a consistent counterexample

64%
Gruenhage G., Koszmider P.
Fundamenta Mathematicae
|
1996
|
tom 149
|
nr 2
143-166
EN
We construct a consistent example of a normal locally compact metacompact space which is not paracompact, answering a question of A. V. Arkhangel'skiĭ and F. Tall. An interplay between a tower in P(ω)/Fin, an almost disjoint family in $[ω]^ω$, and a version of an (ω,1)-morass forms the core of the proof. A part of the poset which forces the counterexample can be considered a modification of a poset due to Judah and Shelah for obtaining a Q-set by a countable support iteration.
3
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Countable Toronto spaces

64%
Gruenhage G., Tach Moore J.
Fundamenta Mathematicae
|
2000
|
tom 163
|
nr 2
143-162
EN
A space X is called an α-Toronto space if X is scattered of Cantor-Bendixson rank α and is homeomorphic to each of its subspaces of the same rank. We answer a question of Steprāns by constructing a countable α-Toronto space for each α ≤ ω. We also construct consistent examples of countable α-Toronto spaces for each $α < ω_1$.
4
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The Arkhangel'skiĭ–Tall problem under Martin’s Axiom

64%
Gruenhage G., Koszmider P.
Fundamenta Mathematicae
|
1996
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tom 149
|
nr 3
275-285
EN
We show that MA$_{σ-centered}(ω_1)$ implies that normal locally compact metacompact spaces are paracompact, and that MA($ω_1$) implies normal locally compact metalindelöf spaces are paracompact. The latter result answers a question of S. Watson. The first result implies that there is a model of set theory in which all normal locally compact metacompact spaces are paracompact, yet there is a normal locally compact metalindelöf space which is not paracompact.
5
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Uniformization and anti-uniformization properties of ladder systems

51%
Eisworth T., Gruenhage G., Pavlov O., Szeptycki P.
Fundamenta Mathematicae
|
2004
|
tom 181
|
nr 3
189-213
EN
Natural weakenings of uniformizability of a ladder system on ω₁ are considered. It is shown that even assuming CH all the properties may be distinct in a strong sense. In addition, these properties are studied in conjunction with other properties inconsistent with full uniformizability, which we call anti-uniformization properties. The most important conjunction considered is the uniformization property we call countable metacompactness and the anti-uniformization property we call thinness. The existence of a thin, countably metacompact ladder system is used to construct interesting topological spaces: a countably paracompact and nonnormal subspace of ω₁², and a countably paracompact, locally compact screenable space which is not paracompact. Whether the existence of a thin, countably metacompact ladder system is consistent is left open. Finally, the relation between the properties introduced and other well known properties of ladder systems, such as ♣, is considered.
6
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Dugundji extenders and retracts on generalized ordered spaces

51%
Gruenhage G., Hattori Y., Ohta H.
Fundamenta Mathematicae
|
1998
|
tom 158
|
nr 2
147-164
EN
For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an $L_{ch}$-extender (resp. $L_{cch}$-extender) if φ(f)[X] is included in the convex hull (resp. closed convex hull) of f[A] for each f ∈ C(A). Consider the following conditions (i)-(vii) for a closed subset A of a GO-space X: (i) A is a retract of X; (ii) A is a retract of the union of A and all clopen convex components of X\A; (iii) there is a continuous $L_{ch}$-extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and the pointwise convergence topology, for each space Y; (iv) A × Y is C*-embedded in X × Y for each space Y; (v) there is a continuous linear extender $φ:C*_{k}(A) → C_{p}(X)$; (vi) there is an $L_{ch}$-extender φ:C(A) → C(X); and (vii) there is an $L_{cch}$-extender φ:C(A) → C(X). We prove that these conditions are related as follows: (i)⇒(ii)⇔(iii)⇔(iv)⇔(v)⇒(vi)⇒(vii). If A is paracompact and the cellularity of A is nonmeasurable, then (ii)-(vii) are equivalent. If there is no connected subset of X which meets distinct convex components of A, then (ii) implies (i). We show that van Douwen's example of a separable GO-space satisfies none of the above conditions, which answers questions of Heath-Lutzer [9], van Douwen [1] and Hattori [8].
7
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On a perfect set theorem of A. H. Stone and N. N. Lusin’s constituents

32%
Chaber J., Gruenhage G., Pol R.
Fundamenta Mathematicae
|
1995
|
tom 148
|
nr 3
309-318
8
Content available remote

On a Corson space of Todorčević

32%
Gruenhage G.
Fundamenta Mathematicae
|
1985-1986
|
tom 126
|
nr 3
261-268
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