Lindelöf indestructibility, topological games and selection principles

Marion Scheepers; Franklin D. Tall

Displaying similar documents to “Lindelöf indestructibility, topological games and selection principles”

Topological games and product spaces

Salvador García-Ferreira, R. A. González-Silva, Artur Hideyuki Tomita (2002)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we deal with the product of spaces which are either $𝒢$ -spaces or $𝒢_{p}$ -spaces, for some $p \in ω^{*}$ . These spaces are defined in terms of a two-person infinite game over a topological space. All countably compact spaces are $𝒢$ -spaces, and every $𝒢_{p}$ -space is a $𝒢$ -space, for every $p \in ω^{*}$ . We prove that if ${X_{μ} : μ < ω_{1}}$ is a set of spaces whose product $X = \prod_{μ < ω_{1}} X_{μ}$ is a $𝒢$ -space, then there is $A \in {[ω_{1}]}^{\leq ω}$ such that $X_{μ}$ is countably compact for every $μ \in ω_{1} ∖ A$ . As a consequence, $X^{ω_{1}}$ is a $𝒢$ -space iff $X^{ω_{1}}$ is countably compact, and if $X^{2^{𝔠}}$ is a $𝒢$ -space,...

Infinite games and chain conditions

Santi Spadaro (2016)

Fundamenta Mathematicae

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We apply the theory of infinite two-person games to two well-known problems in topology: Suslin’s Problem and Arhangel’skii’s problem on the weak Lindelöf number of the $G_{δ}$ topology on a compact space. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable, and 2) in every compact space satisfying the game-theoretic version of the weak...

More on the Ehrenfeucht-Fraisse game of length ω₁

Tapani Hyttinen, Saharon Shelah, Jouko Vaananen (2002)

Fundamenta Mathematicae

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By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, $E F G_{ω ₁} (,)$ , is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and $E F G_{ω ₁} (,)$ is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if $2^{ℵ ₀} < 2^{ℵ ₃}$ , T is a countable...

Infinite $G_{δ}$ -games with imperfect information

D. Blackwell (1969)

Applicationes Mathematicae

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A solution to Comfort's question on the countable compactness of powers of a topological group

Artur Hideyuki Tomita (2005)

Fundamenta Mathematicae

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In 1990, Comfort asked Question 477 in the survey book “Open Problems in Topology”: Is there, for every (not necessarily infinite) cardinal number $α \leq 2^{}$ , a topological group G such that $G^{γ}$ is countably compact for all cardinals γ < α, but $G^{α}$ is not countably compact? Hart and van Mill showed in 1991 that α = 2 answers this question affirmatively under $M A_{c o u n t a b l e}$ . Recently, Tomita showed that every finite cardinal answers Comfort’s question in the affirmative, also from $M A_{c o u n t a b l e}$ . However, the question has...

Spaces with star countable extent

A. D. Rojas-Sánchez, Angel Tamariz-Mascarúa (2016)

Commentationes Mathematicae Universitatis Carolinae

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For a topological property $P$ , we say that a space $X$ is star $P$ if for every open cover $𝒰$ of the space $X$ there exists $A \subset X$ such that $s t (A, 𝒰) = X$ . We consider space with star countable extent establishing the relations between the star countable extent property and the properties star Lindelöf and feebly Lindelöf. We describe some classes of spaces in which the star countable extent property is equivalent to either the Lindelöf property or separability. An example is given of a Tychonoff star Lindelöf...

Nonnormality of remainders of some topological groups

Aleksander V. Arhangel&#039;skii, J. van Mill (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group $G$ has a normal remainder. In a previous paper we showed that under mild conditions on $G$ , the Continuum Hypothesis implies that if the Čech-Stone remainder $G^{*}$ of $G$ is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight...

Combinatorics of open covers (VII): Groupability

Ljubiša D. R. Kočinac, Marion Scheepers (2003)

Fundamenta Mathematicae

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We use Ramseyan partition relations to characterize: ∙ the classical covering property of Hurewicz; ∙ the covering property of Gerlits and Nagy; ∙ the combinatorial cardinal numbers and add(ℳ ). Let X be a $T_{31 / 2}$ -space. In [9] we showed that $C_{p} (X)$ has countable strong fan tightness as well as the Reznichenko property if, and only if, all finite powers of X have the Gerlits-Nagy covering property. Now we show that the following are equivalent: 1. $C_{p} (X)$ has countable fan tightness and the Reznichenko...

On topological groups with a small base and metrizability

Saak Gabriyelyan, Jerzy Kąkol, Arkady Leiderman (2015)

Fundamenta Mathematicae

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A (Hausdorff) topological group is said to have a -base if it admits a base of neighbourhoods of the unit, $U_{α} : α \in ℕ^{ℕ}$ , such that $U_{α} \subset U_{β}$ whenever β ≤ α for all $α, β \in ℕ^{ℕ}$ . The class of all metrizable topological groups is a proper subclass of the class $T G_{}$ of all topological groups having a -base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a -base. We also show that any precompact set in a topological group $G \in T G_{}$ is metrizable, and hence G is strictly angelic. We deduce from...

A note on star Lindelöf, first countable and normal spaces

Wei-Feng Xuan (2017)

Mathematica Bohemica

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A topological space $X$ is said to be star Lindelöf if for any open cover $𝒰$ of $X$ there is a Lindelöf subspace $A \subset X$ such that $St (A, 𝒰) = X$ . The “extent” $e (X)$ of $X$ is the supremum of the cardinalities of closed discrete subsets of $X$ . We prove that under $V = L$ every star Lindelöf, first countable and normal space must have countable extent. We also obtain an example under $MA + \neg CH$ , which shows that a star Lindelöf, first countable and normal space may not have countable extent.

On the Variational Inequality and Tykhonov Well-Posedness in Game Theory

C. A. Pensavalle, G. Pieri (2010)

Bollettino dell'Unione Matematica Italiana

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Consider a M-player game in strategic form $G=(X_{1},\cdots,X_{M},g_{1},\cdots,g_{M})$ where the set $X_{i}$ is a closed interval of real numbers and the payoff function $g_{i}$ is concave and differentiable with respect to the variable $x_{i}\in X_{i}$ , for any $i=1,\cdots,M$ . The aim of this paper is to find appropriate conditions on the payoff functions under the well-posedness with respect to the related variational inequality is equivalent to the formulation of the Tykhonov well-posedness in a game context. The idea of the proof is to appeal to a third equivalence,...

Some applications of the point-open subbase game

D. Guerrero Sánchez, Vladimir Vladimirovich Tkachuk (2017)

Commentationes Mathematicae Universitatis Carolinae

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Given a subbase $𝒮$ of a space $X$ , the game $P O (𝒮, X)$ is defined for two players $P$ and $O$ who respectively pick, at the $n$ -th move, a point $x_{n} \in X$ and a set $U_{n} \in 𝒮$ such that $x_{n} \in U_{n}$ . The game stops after the moves ${x_{n}, U_{n} : n \in ø}$ have been made and the player $P$ wins if $⋃_{n \in ø} U_{n} = X$ ; otherwise $O$ is the winner. Since $P O (𝒮, X)$ is an evident modification of the well-known point-open game $P O (X)$ , the primary line of research is to describe the relationship between $P O (X)$ and $P O (𝒮, X)$ for a given subbase $𝒮$ . It turns out that, for any subbase $𝒮$ , the player $P$ has a winning...

On star covering properties related to countable compactness and pseudocompactness

Marcelo D. Passos, Heides L. Santana, Samuel G. da Silva (2017)

Commentationes Mathematicae Universitatis Carolinae

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We prove a number of results on star covering properties which may be regarded as either generalizations or specializations of topological properties related to the ones mentioned in the title of the paper. For instance, we give a new, entirely combinatorial proof of the fact that $Ψ$ -spaces constructed from infinite almost disjoint families are not star-compact. By going a little further we conclude that if $X$ is a star-compact space within a certain class, then $X$ is neither first countable...

A countably cellular topological group all of whose countable subsets are closed need not be $ℝ$ -factorizable

Mihail G. Tkachenko (2023)

Commentationes Mathematicae Universitatis Carolinae

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We construct a Hausdorff topological group $G$ such that $ℵ_{1}$ is a precalibre of $G$ (hence, $G$ has countable cellularity), all countable subsets of $G$ are closed and $C$ -embedded in $G$ , but $G$ is not $ℝ$ -factorizable. This solves Problem 8.6.3 from the book “Topological Groups and Related Structures" (2008) in the negative.

Remainders of metrizable spaces and a generalization of Lindelöf Σ-spaces

A. V. Arhangel'skii (2011)

Fundamenta Mathematicae

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We establish some new properties of remainders of metrizable spaces. In particular, we show that if the weight of a metrizable space X does not exceed $2^{ω}$ , then any remainder of X in a Hausdorff compactification is a Lindelöf Σ-space. An example of a metrizable space whose remainder in some compactification is not a Lindelöf Σ-space is given. A new class of topological spaces naturally extending the class of Lindelöf Σ-spaces is introduced and studied. This leads to the following theorem:...

Characterizations of $z$ -Lindelöf spaces

Ahmad Al-Omari, Takashi Noiri (2017)

Archivum Mathematicum

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A topological space $(X, τ)$ is said to be $z$ -Lindelöf [1] if every cover of $X$ by cozero sets of $(X, τ)$ admits a countable subcover. In this paper, we obtain new characterizations and preservation theorems of $z$ -Lindelöf spaces.

On universality of countable and weak products of sigma hereditarily disconnected spaces

Taras Banakh, Robert Cauty (2001)

Fundamenta Mathematicae

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Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power $X^{ω}$ of any subspace X ⊂ Y is not universal for the class ₂ of absolute $G_{δ σ}$ -sets; moreover, if Y is an absolute $F_{σ δ}$ -set, then $X^{ω}$ contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute $G_{δ}$ -set, then $X^{ω}$ contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable...

The (dis)connectedness of products of Hausdorff spaces in the box topology

Vitalij A. Chatyrko (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper the following two propositions are proved: (a) If $X_{α}$ , $α \in A$ , is an infinite system of connected spaces such that infinitely many of them are nondegenerated completely Hausdorff topological spaces then the box product $□_{α \in A} X_{α}$ can be decomposed into continuum many disjoint nonempty open subsets, in particular, it is disconnected. (b) If $X_{α}$ , $α \in A$ , is an infinite system of Brown Hausdorff topological spaces then the box product $□_{α \in A} X_{α}$ is also Brown Hausdorff, and hence, it is connected. A space...