Currently displaying 1 – 14 of 14

Showing per page

Order by Relevance | Title | Year of publication

Partitions of compact Hausdorff spaces

Gary Gruenhage — 1993

Fundamenta Mathematicae

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.

The Arkhangel’skiĭ–Tall problem: a consistent counterexample

Gary GruenhagePiotr Koszmider — 1996

Fundamenta Mathematicae

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.

The Arkhangel'skiĭ–Tall problem under Martin’s Axiom

Gary GruenhagePiotr Koszmider — 1996

Fundamenta Mathematicae

We show that MA σ - c e n t e r e d ( 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.

Countable Toronto spaces

Gary GruenhageJ. Moore — 2000

Fundamenta Mathematicae

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 .

Covering ω ω by special Cantor sets

Gary GruenhageRonnie Levy — 2002

Commentationes Mathematicae Universitatis Carolinae

This paper deals with questions of how many compact subsets of certain kinds it takes to cover the space ω ω of irrationals, or certain of its subspaces. In particular, given f ω ( ω { 0 } ) , we consider compact sets of the form i ω B i , where | B i | = f ( i ) for all, or for infinitely many, i . We also consider “ n -splitting” compact sets, i.e., compact sets K such that for any f K and i ω , | { g ( i ) : g K , g i = f i } | = n .

A game and its relation to netweight and D-spaces

Gary GruenhagePaul Szeptycki — 2011

Commentationes Mathematicae Universitatis Carolinae

We introduce a two player topological game and study the relationship of the existence of winning strategies to base properties and covering properties of the underlying space. The existence of a winning strategy for one of the players is conjectured to be equivalent to the space have countable network weight. In addition, connections to the class of D-spaces and the class of hereditarily Lindelöf spaces are shown.

Dugundji extenders and retracts on generalized ordered spaces

Gary GruenhageYasunao HattoriHaruto Ohta — 1998

Fundamenta Mathematicae

For a subspace A of a space X, a linear extender φ:C(A) → C(X) is called an L c h -extender (resp. L c c h -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; (iii) there is a continuous L c h -extender φ:C(A × Y) → C(X × Y), with respect to both the compact-open topology and...

Uniformization and anti-uniformization properties of ladder systems

Todd EisworthGary GruenhageOleg PavlovPaul Szeptycki — 2004

Fundamenta Mathematicae

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...

Page 1

Download Results (CSV)