Selections and suborderability

Artico, Giuliano; Marconi, U.; Pelant, Jan; Rotter, Luca; Tkachenko, Mikhail

doi:10.4064/fm175-1-1

Cited by 41 publications

(40 citation statements)

References 20 publications

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“…In this paper, we show that countable compactness can be replaced by pseudocompactness. Our main result solves Problems 5.1 and 5.2 from [1]. J. van Mill and E. Wattel [4] proved that a compact space X has a weak selection iff it is orderable.…”

Section: Introductionmentioning

confidence: 81%

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Weak selections and pseudocompactness

Garcı́a-Ferreira¹,

Sanchis

2004

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 81%

“…A direct consequence of Theorem 2.3 and the Theorems from [1] and [5] which were mentioned in the Introduction is the following. Corollary 2.4.…”

Section: S Garcia-ferreira and M Sanchismentioning

confidence: 98%

Weak selections and pseudocompactness

Garcı́a-Ferreira¹,

Sanchis

2004

Proc. Amer. Math. Soc.

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“…Every continuous weak selection f for X can be considered as a continuous map f : X × X → X such that f (x, y) = f (y, x) and f (x, y) ∈ {x, y} for every x, y ∈ X, see [1]. Another way is to look at f as the relation f on X which is both total and antisymmetric, i.e.…”

Section: Totally Disconnected Spaces and Weakly Extreme Pointsmentioning

confidence: 99%

“…There is a space X with se [F (X)] = ∅, and a selection minimal point p ∈ X which is a cut point of X but X is not zero-dimensional at p. P r o o f. Let C be the Cantor set in the interval [0, 1], p = 1 ∈ C, ≤ be the linear ordering on C as a subset of [0,1], and let D = {d n : n < ω} ⊂ C be a strictly increasing sequence convergent to p. Define another topology on C in which a set…”

Section: Totally Disconnected Spaces and Weakly Extreme Pointsmentioning

confidence: 99%

Selections and countable compactness

Buhagiar

Gutev

2013

Mathematica Slovaca

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“…We refer the interested reader to [9] for a detailed survey of connections of selective pseudocompactness and strong pseudocompactness to other classical compactness-like notions, including the notion of sequential pseudocompactness of Artico, Marconi, Pelant, Rotter and Tkachenko [1].…”

Section: Introductionmentioning

confidence: 99%

Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

Dorantes-Aldama

Shakhmatov

2017

Topology and its Applications

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Abstract. A space X is selectively sequentially pseudocompact if for every family {Un : n ∈ N} of non-empty open subsets of X, one can choose a point xn ∈ Un for every n ∈ N in such a way that the sequence {xn : n ∈ N} has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of García-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of García-Ferreira and Tomita.The symbol N denotes the set of natural numbers and N + = N \ {0} denotes the set of positive integer numbers. The symbol Z denotes the group of integer numbers, R the set of real numbers and T the circle group {e iθ : θ ∈ R} ⊆ R 2 . The symbols ω, ω 1 and c stand for the first infinite cardinal, the first uncountable cardinal and the cardinality of the continuum, respectively. For a set X, the set of all finite subsets of X is denoted by [X] <ω , while [X] ω denotes the set of all countably infinite subsets of X.If X is a subset of a group G, then X is the smallest subgroup of G that contains X. For groups which are not necessary Abelian we use the multiplication notation, while for Abelian groups we always use the additive one. In particular, e denotes the identity element of a group and 0 is used for the zero element of an Abelian group. Recall that an element g of a group G is torsion if g n = e for some positive integer n. A group is torsion if all of its elements are torsion. A group G is bounded torsion if there exists n ∈ N such that g n = e for all g ∈ G.Let G be an Abelian group. The symbol t(G) stands for the subgroup of torsion elements of G. For each n ∈ N, note that nG = {ng : g ∈ G} is a subgroup of G and the map g → ng (g ∈ G) is a homomorphism of G onto nG. For a cardinal τ we denote by G (τ ) the direct sum of τ copies of the group G.We will say that a sequence {x n : n ∈ N} of points in a topological space X:• converges to a point x ∈ X if the set {n ∈ N : x n ∈ W } is finite for every open neighbourhood W of x in X; • is convergent if it converges to some point of X.

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Selections and suborderability

Cited by 41 publications

References 20 publications

Weak selections and pseudocompactness

Weak selections and pseudocompactness

Selections and countable compactness

Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

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