# On weakening tightness to weak tightness

@article{Bella2019OnWT,
title={On weakening tightness to weak tightness},
author={Angelo Bella and Nathan A. Carlson},
journal={Monatshefte f{\"u}r Mathematik},
year={2019},
volume={192},
pages={39-48}
}
• Published 15 January 2019
• Mathematics, Physics
• Monatshefte für Mathematik
The weak tightness wt ( X ) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018 ) with the property $$wt(X)\le t(X)$$ w t ( X ) ≤ t ( X ) . We investigate several well-known results concerning t ( X ) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that $$wt(X)=\aleph _0<t(X)$$ w t ( X ) = ℵ 0 < t ( X ) under $$2^{\aleph _0}=2^{\aleph _1}$$ 2 ℵ 0 = 2 ℵ 1 . In particular, this demonstrates the…
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#### References

SHOWING 1-10 OF 31 REFERENCES
On the weak tightness and power homogeneous compacta
• N. Carlson
• Mathematics
Topology and its Applications
• 2018
Abstract Motivated by results of Juhasz and van Mill in [13] , we define the cardinal invariant w t ( X ) , the weak tightness of a topological space X , and show that | X | ≤ 2 L ( X ) w t ( X ) ψ (
On $\sigma$-countably tight spaces
• Mathematics
• 2016
Extending a result of R. de la Vega, we prove that an infinite homogeneous compactum has cardinality $\mathfrak{c}$ if either it is the union of countably many dense or finitely many arbitrary
Observations on some cardinality bounds
Abstract We try to find a common extension of two cardinal inequalities for Lindelof spaces. Using an estimate of the number of G δ points due to Balogh, we improve a result of Juhasz and Spadaro. A
A new bound on the cardinality of homogeneous compacta
Abstract We show (in ZFC) that if X is a compact homogeneous Hausdorff space then | X | ⩽ 2 t ( X ) , where t ( X ) denotes the tightness of X . It follows that under GCH the character and the
A new bound on the cardinality of power homogeneous compacta
• Mathematics
• 2007
It was recently proved by R. de la Vega that if X is a homogeneous compactum then the cardinality of X is bounded by 2t(X), where t(X) denotes the tightness of X. We extend de la Vega's argument to
STRUCTURE AND CLASSIFICATION OF TOPOLOGICAL SPACES AND CARDINAL INVARIANTS
CONTENTSIntroduction Some special terminology and notation Chapter I. Cardinal invariants in broad classes of spaces § 1. Basic cardinal invariants. Typical relations and problems § 2. Cardinal
Projective σ-compactness, ω1-caliber, and Cp-spaces
Abstract A space is called projectively σ -compact, if every separable metrizable continuous image of this space is σ -compact. In particular, we establish when C p ( X ) is projectively σ -compact.
The Souslin number in set-theoretic topology
Abstract This paper presents a short survey of some directions in the theory of cardinal functions, in which estimations containing the Souslin number play the central role. Together with the known
On Compact Hausdorff Spaces of Countable Tightness
• Mathematics
• 1989
A general combinatorial theorem for countably compact, noncompact spaces is given under the Proper Forcing Axiom. It follows that compact Hausdorff spaces of countable tightness are sequential under
On the cardinality of power homogeneous Hausdorff spaces
We prove that the cardinality of power homogeneous Hausdorff spaces X is bounded by d(X) ��(X) . This inequality improves many known results and it also solves a question by J. van Mill. We further