Lindelöf property and absolute embeddings

Bella, Angelo; Yaschenko, Ivan

doi:10.1090/s0002-9939-99-04568-2

Cited by 15 publications

(21 citation statements)

References 6 publications

(4 reference statements)

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“…In particular, A. V. Arhangel'skii and D. N. Stavrova characterized these spaces [1,4] and A. Bella, V. V. Tkachuk and I. V. Yaschenko discussed the relationships among these spaces and AP or WAP spaces [5,6,18].…”

Section: Wc Hong and S Kwonmentioning

confidence: 99%

“…Recall that a space X is AP (standing for Approximation by Points) [6,9,15] (also called Whyburn [16]…”

Section: It Is Obvious That For Each Subsetmentioning

confidence: 99%

“…X is WAP (standing for Weak Approximation by Points) [6,9,15](also called weakly Whyburn [16]) iff for each subset A of X, A = [A] AP implies A = A. X is WACP (standing for Weak Approximation by Countable Points) [9] iff for each subset A of X,…”

Section: It Is Obvious That For Each Subsetmentioning

confidence: 99%

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A Note on Spaces Determined by Closure-Like Operators

Hong¹,

Kwon²

2016

East Asian mathematical journal

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Abstract. In this paper, we study some classes of spaces determined by closure-like operators [·]s, [·]c and [·] k etc. which are wider than the class of Fréchet-Urysohn spaces or the class of sequential spaces and related spaces. We first introduce a WADS space which is a generalization of a sequential space. We show that X is a WADS and k-space iff X is sequential and every WADS space is C-closed and obtained that every WADS and countably compact space is sequential as a corollary. We also show that every WAP and countably compact space is countably sequential and obtain that every WACP and countably compact space is sequential as a corollary. And we show that every WAP and weakly kspace is countably sequential and obtain that X is a WACP and weakly k-space iff X is sequential as a corollary.

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Section: Wc Hong and S Kwonmentioning

confidence: 99%

“…Recall that a space X is AP (standing for Approximation by Points) [6,9,15] (also called Whyburn [16]…”

Section: It Is Obvious That For Each Subsetmentioning

confidence: 99%

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A Note on Spaces Determined by Closure-Like Operators

Hong¹,

Kwon²

2016

East Asian mathematical journal

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“…A space X is countably Fréchet-Urysohn [8] iff for each countable subset A of X and each x ∈ A, there exists a sequence (x n ) of points of A such that (x n ) converges to x in X. A space X is AP (standing for Approximation by Points) [3] (also called Whyburn [11]) iff for each non-closed subset A of X and each x ∈ A \ A, there exists a subset B of A such that B = B ∪ {x}. A space X is WAP (standing for Weak Approximation by Points) [2] (also called weakly Whyburn [11]) iff for each subset A of X which is not closed in X, there exist x ∈ A \ A and a subset B of A such that B = B ∪ {x}.…”

Section: Introductionmentioning

confidence: 99%

“…Several authors (see [1,2,3,4,6,8,10,11,13]) studied some properties of a sequential space and relations between a sequential space and related spaces. In particular, in [6,7,8,9,13], the authors showed some sufficient conditions for a sequential space to be Fréchet-Urysohn.…”

Section: Introductionmentioning

confidence: 99%

A Generalization of a Sequential Space and Related Spaces

Hong¹,

Kwon²

2014

Honam Mathematical Journal

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Variations of selective separability II: Discrete sets and the influence of convergence and maximality

Bella

Matveev²,

Spadaro

2012

Topology and its Applications

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A space X is called selectively separable (R-separable) if for every sequence of dense subspaces (Dn : n ∈ ω) one can pick finite (respectively, onepoint) subsets Fn ⊂ Dn such that n∈ω Fn is dense in X. These properties are much stronger than separability, but are equivalent to it in the presence of certain convergence properties. For example, we show that every Hausdorff separable radial space is R-separable and note that neither separable sequential nor separable Whyburn spaces have to be selectively separable. A space is called d-separable if it has a dense σ-discrete subspace. We call a space X D-separable if for every sequence of dense subspaces (Dn : n ∈ ω) one can pick discrete subsets Fn ⊂ Dn such that n∈ω Fn is dense in X. Although dseparable spaces are often also D-separable (this is the case, for example, with linearly ordered d-separable or stratifiable spaces), we offer three examples of countable non-D-separable spaces. It is known that d-separability is preserved by arbitrary products, and that for every X, the power X d(X) is d-separable. We show that D-separability is not preserved even by finite products, and that for every infinite X, the power X 2 d(X) is not D-separable. However, for every X there is a Y such that X × Y is D-separable. Finally, we discuss selective and D-separability in the presence of maximality. For example, we show that (assuming d = c) there exists a maximal regular countable selectively separable space, and that (in ZFC) every maximal countable space is D-separable (while some of those are not selectively separable). However, no maximal space satisfies the natural game-theoretic strengthening of D-separability.2000 Mathematics Subject Classification. 54D65, 54A25, 54D55, 54A20.

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Lindelöf property and absolute embeddings

Cited by 15 publications

References 6 publications

A Note on Spaces Determined by Closure-Like Operators

A Note on Spaces Determined by Closure-Like Operators

A Generalization of a Sequential Space and Related Spaces

Variations of selective separability II: Discrete sets and the influence of convergence and maximality

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