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

• Autorzy: Harold Bennett , David Lutzer
• Języki publikacji: EN

Abstrakty

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.

Kategorie tematyczne

• 54E52: Baire category, Baire spaces
• 54D30: Compactness
• 54F05: Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
• 46B03: Isomorphic theory (including renorming) of Banach spaces
• 46B50: Compactness in Banach (or normed) spaces
• 46B26: Nonseparable Banach spaces
• 46A99: None of the above, but in this section

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2013
• Tom: 223
• Numer: 3
• Strony: 273-294

Daty

wydano

2013

Twórcy

autor

Harold Bennett

• Texas Tech University, Lubbock, TX 79405, U.S.A.

autor

David Lutzer

• College of William and Mary, Williamsburg, Va 23187, U.S.A.

Identyfikatory

DOI

10.4064/fm223-3-4