Currently displaying 1 – 10 of 10

Showing per page

Order by Relevance | Title | Year of publication

Almost disjoint families and property (a)

Paul SzeptyckiJerry Vaughan — 1998

Fundamenta Mathematicae

We consider the question: when does a Ψ-space satisfy property (a)? We show that if | A | < p then the Ψ-space Ψ(A) satisfies property (a), but in some Cohen models the negation of CH holds and every uncountable Ψ-space fails to satisfy property (a). We also show that in a model of Fleissner and Miller there exists a Ψ-space of cardinality p which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).

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.

Gδ -sets in topological spaces and games

Winfried JustMarion ScheepersJuris StepransPaul Szeptycki — 1997

Fundamenta Mathematicae

Players ONE and TWO play the following game: In the nth inning ONE chooses a set O n from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset T n of X. The players must obey the rule that O n O n + 1 T n + 1 T n for each n. TWO wins if the intersection of TWO’s sets is equal to the union of ONE’s sets. If ONE has no winning strategy, then each element of ℱ is a G δ -set. To what extent is the converse true? We show that:  (A) For ℱ the collection of countable subsets of X:   1. There are subsets...

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...

Wijsman hyperspaces of non-separable metric spaces

Rodrigo Hernández-GutiérrezPaul J. Szeptycki — 2015

Fundamenta Mathematicae

Given a metric space ⟨X,ρ⟩, consider its hyperspace of closed sets CL(X) with the Wijsman topology τ W ( ρ ) . It is known that C L ( X ) , τ W ( ρ ) is metrizable if and only if X is separable, and it is an open question by Di Maio and Meccariello whether this is equivalent to C L ( X ) , τ W ( ρ ) being normal. We prove that if the weight of X is a regular uncountable cardinal and X is locally separable, then C L ( X ) , τ W ( ρ ) is not normal. We also solve some questions by Cao, Junnila and Moors regarding isolated points in Wijsman hyperspaces.

MAD families and P -points

Salvador García-FerreiraPaul J. Szeptycki — 2007

Commentationes Mathematicae Universitatis Carolinae

The Katětov ordering of two maximal almost disjoint (MAD) families 𝒜 and is defined as follows: We say that 𝒜 K if there is a function f : ω ω such that f - 1 ( A ) ( ) for every A ( 𝒜 ) . In [Garcia-Ferreira S., Hrušák M., Ordering MAD families a la Katětov, J. Symbolic Logic 68 (2003), 1337–1353] a MAD family is called K -uniform if for every X ( 𝒜 ) + , we have that 𝒜 | X K 𝒜 . We prove that CH implies that for every K -uniform MAD family 𝒜 there is a P -point p of ω * such that the set of all Rudin-Keisler predecessors of p is dense in the...

Selections on Ψ -spaces

Michael HrušákPaul J. SzeptyckiArtur Hideyuki Tomita — 2001

Commentationes Mathematicae Universitatis Carolinae

We show that if 𝒜 is an uncountable AD (almost disjoint) family of subsets of ω then the space Ψ ( 𝒜 ) does not admit a continuous selection; moreover, if 𝒜 is maximal then Ψ ( 𝒜 ) does not even admit a continuous selection on pairs, answering thus questions of T. Nogura.

Page 1

Download Results (CSV)