On the cardinality of the θ-closed hull of sets

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

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5 Scopus citations


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.

Original languageEnglish
Pages (from-to)2371-2378
Number of pages8
JournalTopology and its Applications
Issue number18
StatePublished - 1 Dec 2013

Bibliographical 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.


  • 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


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