Wyniki wyszukiwania

1

Domain-representable spaces

100%

Bennett H., Lutzer D.

Fundamenta Mathematicae

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2006

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tom 189

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nr 3

255-268

EN

We study domain-representable spaces, i.e., spaces that can be represented as the space of maximal elements of some continuous directed-complete partial order (= domain) with the Scott topology. We show that the Michael and Sorgenfrey lines are of this type, as is any subspace of any space of ordinals. We show that any completely regular space is a closed subset of some domain-representable space, and that if X is domain-representable, then so is any $G_{δ}$-subspace of X. It follows that any Čech-complete space is domain-representable. These results answer several questions in the literature.

2

The Gruenhage property, property *, fragmentability, and σ-isolated networks in generalized ordered spaces

100%

Bennett H., Lutzer D.

Fundamenta Mathematicae

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2013

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tom 223

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nr 3

273-294

EN

We examine the Gruenhage property, property * (introduced by Orihuela, Smith, and Troyanski), fragmentability, and the existence of σ-isolated networks in the context of linearly ordered topological spaces (LOTS), generalized ordered spaces (GO-spaces), and monotonically normal spaces. We show that any monotonically normal space with property * or with a σ-isolated network must be hereditarily paracompact, so that property * and the Gruenhage property are equivalent in monotonically normal spaces. (However, a fragmentable monotonically normal space may fail to be paracompact.) We show that any fragmentable GO-space must have a σ-disjoint π-base and it follows from a theorem of H. E. White that any fragmentable, first-countable GO-space has a dense metrizable subspace. We also show that any GO-space that is fragmentable and is a Baire space has a dense metrizable subspace. We show that in any compact LOTS X, metrizability is equivalent to each of the following: X is Eberlein compact; X is Talagrand compact; X is Gulko compact; X has a σ-isolated network; X is a Gruenhage space; X has property *; X is perfect and fragmentable; the function space C(X)* has a strictly convex dual norm. We give an example of a GO-space that has property *, is fragmentable, and has a σ-isolated network and a σ-disjoint π-base but contains no dense metrizable subspace.

3

Domain representability of $C_{p}(X)$

100%

Bennett H., Lutzer D.

Fundamenta Mathematicae

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2008

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tom 200

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nr 2

185-199

EN

Let $C_{p}(X)$ be the space of continuous real-valued functions on X, with the topology of pointwise convergence. We consider the following three properties of a space X: (a) $C_{p}(X)$ is Scott-domain representable; (b) $C_{p}(X)$ is domain representable; (c) X is discrete. We show that those three properties are mutually equivalent in any normal T₁-space, and that properties (a) and (c) are equivalent in any completely regular pseudo-normal space. For normal spaces, this generalizes the recent result of Tkachuk that $C_{p}(X)$ is subcompact if and only if X is discrete.

4

Some questions of Arhangel'skii on rotoids

81%

Bennett H., Burke D., Lutzer D.

Fundamenta Mathematicae

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2012

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tom 216

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nr 2

147-161

EN

A rotoid is a space X with a special point e ∈ X and a homeomorphism F: X² → X² having F(x,x) = (x,e) and F(e,x) = (e,x) for every x ∈ X. If any point of X can be used as the point e, then X is called a strong rotoid. We study some general properties of rotoids and prove that the Sorgenfrey line is a strong rotoid, thereby answering several questions posed by A. V. Arhangel'skii, and we pose further questions.

Ograniczanie wyników

4 Fundamenta Mathematicae

4 Bennett H.

4 Lutzer D.

1 Burke D.

1 2013

1 2012

1 2008

1 2006

Domain-representable spaces

The Gruenhage property, property *, fragmentability, and σ-isolated networks in generalized ordered spaces

Domain representability of $C_{p}(X)$

Some questions of Arhangel'skii on rotoids