Displaying similar documents to “Closure-preserving covers in function spaces”

A game and its relation to netweight and D-spaces

Gary Gruenhage, Paul Szeptycki (2011)

Commentationes Mathematicae Universitatis Carolinae

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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.

On the Lindelöf property of spaces of continuous functions over a Tychonoff space and its subspaces

Oleg Okunev (2009)

Commentationes Mathematicae Universitatis Carolinae

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We study relations between the Lindelöf property in the spaces of continuous functions with the topology of pointwise convergence over a Tychonoff space and over its subspaces. We prove, in particular, the following: a) if C p ( X ) is Lindelöf, Y = X ∪ { p } , and the point p has countable character in Y , then C p ( Y ) is Lindelöf; b) if Y is a cozero subspace of a Tychonoff space X , then l ( C p ( Y ) ω ) ≤ l ( C p ( X ) ω ) and ext ( C p ( Y ) ω ) ≤ ext ( C p ( X ) ω ) .

A nice class extracted from C p -theory

Vladimir Vladimirovich Tkachuk (2005)

Commentationes Mathematicae Universitatis Carolinae

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We study systematically a class of spaces introduced by Sokolov and call them Sokolov spaces. Their importance can be seen from the fact that every Corson compact space is a Sokolov space. We show that every Sokolov space is collectionwise normal, ω -stable and ω -monolithic. It is also established that any Sokolov compact space X is Fréchet-Urysohn and the space C p ( X ) is Lindelöf. We prove that any Sokolov space with a G δ -diagonal has a countable network and obtain some cardinality restrictions...

On the open-open game

Peg Daniels, Kenneth Kunen, Haoxuan Zhou (1994)

Fundamenta Mathematicae

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We modify a game due to Berner and Juhász to get what we call “the open-open game (of length ω)”: a round consists of player I choosing a nonempty open subset of a space X and II choosing a nonempty open subset of I’s choice; I wins if the union of II’s open sets is dense in X, otherwise II wins. This game is of interest for ccc spaces. It can be translated into a game on partial orders (trees and Boolean algebras, for example). We present basic results and various conditions under which...

On β-favorability of the strong Choquet game

László Zsilinszky (2011)

Colloquium Mathematicae

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In the main result, partially answering a question of Telgársky, the following is proven: if X is a first countable R₀-space, then player β (i.e. the EMPTY player) has a winning strategy in the strong Choquet game on X if and only if X contains a nonempty W δ -subspace which is of the first category in itself.

A dichotomy on Schreier sets

Robert Judd (1999)

Studia Mathematica

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We show that the Schreier sets S α ( α < ω 1 ) have the following dichotomy property. For every hereditary collection ℱ of finite subsets of ℱ, either there exists infinite M = ( m i ) i = 1 ∞ ⊆ ℕ such that S α ( M ) = m i : i ∈ E : E ∈ S α ⊆ ℱ , or there exist infinite M = ( m i ) i = 1 ∞ , N ⊆ ℕ such that ℱ [ N ] ( M ) = m i : i ∈ F : F ∈ ℱ a n d F ⊂ N ⊆ S α .