On the cardinality of the θ-closed hull of sets

Filippo Cammaroto; Andrei Catalioto; Bruno Antonio Pansera; Boaz Tsaban

doi:10.1016/j.topol.2013.07.031

On the cardinality of the θ-closed hull of sets

Filippo Cammaroto, Andrei Catalioto, Bruno Antonio Pansera, Boaz Tsaban

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all closed neighborhoods of a point intersect C, this point is in C.We define a new topological cardinal invariant function, the θ-bitightness small number of a space X, bts_θ(X), and prove that in every topological space X, the cardinality of the θ-closed hull of each set A is at most |A|^btsθ(X). Using this result, we synthesize all earlier results on bounds on the cardinality of θ-closed hulls. We provide applications to P-spaces and to the almost-Lindelöf number.

Original language	English
Pages (from-to)	2371-2378
Number of pages	8
Journal	Topology and its Applications
Volume	160
Issue number	18
DOIs	https://doi.org/10.1016/j.topol.2013.07.031
State	Published - 1 Dec 2013

Keywords

Cardinal inequalities
Character
Finite θ-bitightness
Finitely-Urysohn space
H-closed space
H-set
N-Urysohn space
Urysohn number
Urysohn space
θ-Bitightness
θ-Bitightness small number
θ-Character
θ-Closed hull
θ-Closure
θ-Tightness

All Science Journal Classification (ASJC) codes

Geometry and Topology

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10.1016/j.topol.2013.07.031

Cite this

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title = "On the cardinality of the θ-closed hull of sets",

abstract = "The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all closed neighborhoods of a point intersect C, this point is in C.We define a new topological cardinal invariant function, the θ-bitightness small number of a space X, btsθ(X), and prove that in every topological space X, the cardinality of the θ-closed hull of each set A is at most |A|btsθ(X). Using this result, we synthesize all earlier results on bounds on the cardinality of θ-closed hulls. We provide applications to P-spaces and to the almost-Lindel{\"o}f number.",

keywords = "Cardinal inequalities, Character, Finite θ-bitightness, Finitely-Urysohn space, H-closed space, H-set, N-Urysohn space, Urysohn number, Urysohn space, θ-Bitightness, θ-Bitightness small number, θ-Character, θ-Closed hull, θ-Closure, θ-Tightness",

author = "Filippo Cammaroto and Andrei Catalioto and Pansera, {Bruno Antonio} and Boaz Tsaban",

note = "Funding Information: This research was partially supported by CNR (GNSAGA) and MIUR , Italy, through “Fondi 40%”. A part of this work was carried out during a visit of the fourth named author at the University of Messina. This author thanks his hosts for their warm hospitality and stimulating atmosphere.",

year = "2013",

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doi = "10.1016/j.topol.2013.07.031",

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AU - Cammaroto, Filippo

AU - Catalioto, Andrei

AU - Pansera, Bruno Antonio

AU - Tsaban, Boaz

N1 - Funding Information: This research was partially supported by CNR (GNSAGA) and MIUR , Italy, through “Fondi 40%”. A part of this work was carried out during a visit of the fourth named author at the University of Messina. This author thanks his hosts for their warm hospitality and stimulating atmosphere.

PY - 2013/12/1

Y1 - 2013/12/1

N2 - The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all closed neighborhoods of a point intersect C, this point is in C.We define a new topological cardinal invariant function, the θ-bitightness small number of a space X, btsθ(X), and prove that in every topological space X, the cardinality of the θ-closed hull of each set A is at most |A|btsθ(X). Using this result, we synthesize all earlier results on bounds on the cardinality of θ-closed hulls. We provide applications to P-spaces and to the almost-Lindelöf number.

AB - The θ-closed hull of a set A in a topological space is the smallest set C containing A such that, whenever all closed neighborhoods of a point intersect C, this point is in C.We define a new topological cardinal invariant function, the θ-bitightness small number of a space X, btsθ(X), and prove that in every topological space X, the cardinality of the θ-closed hull of each set A is at most |A|btsθ(X). Using this result, we synthesize all earlier results on bounds on the cardinality of θ-closed hulls. We provide applications to P-spaces and to the almost-Lindelöf number.

KW - Cardinal inequalities

KW - Character

KW - Finite θ-bitightness

KW - Finitely-Urysohn space

KW - H-closed space

KW - H-set

KW - N-Urysohn space

KW - Urysohn number

KW - Urysohn space

KW - θ-Bitightness

KW - θ-Bitightness small number

KW - θ-Character

KW - θ-Closed hull

KW - θ-Closure

KW - θ-Tightness

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U2 - 10.1016/j.topol.2013.07.031

DO - 10.1016/j.topol.2013.07.031

M3 - مقالة

SN - 0166-8641

VL - 160

SP - 2371

EP - 2378

JO - Topology and its Applications

JF - Topology and its Applications

IS - 18

ER -

On the cardinality of the θ-closed hull of sets

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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