On the cardinality of Hausdorff spaces and Pol-Šapirovskii technique

Alejandro Ramírez-Páramo

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 1, page 131-135
  • ISSN: 0010-2628

Abstract

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In this paper we make use of the Pol-Šapirovskii technique to prove three cardinal inequalities. The first two results are due to Fedeli [2] and the third theorem of this paper is a common generalization to: (a) (Arhangel’skii [1]) If X is a T 1 space such that (i) L ( X ) t ( X ) κ , (ii) ψ ( X ) 2 κ , and (iii) for all A [ X ] 2 κ , A ¯ 2 κ , then | X | 2 κ ; and (b) (Fedeli [2]) If X is a T 2 -space then | X | 2 aql ( X ) t ( X ) ψ c ( X ) .

How to cite

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Ramírez-Páramo, Alejandro. "On the cardinality of Hausdorff spaces and Pol-Šapirovskii technique." Commentationes Mathematicae Universitatis Carolinae 46.1 (2005): 131-135. <http://eudml.org/doc/249542>.

@article{Ramírez2005,
abstract = {In this paper we make use of the Pol-Šapirovskii technique to prove three cardinal inequalities. The first two results are due to Fedeli [2] and the third theorem of this paper is a common generalization to: (a) (Arhangel’skii [1]) If $X$ is a $T_\{1\}$ space such that (i) $L(X)t(X)\le \kappa $, (ii) $\psi (X)\le 2^\{\kappa \}$, and (iii) for all $A \in [X]^\{\le 2^\{\kappa \}\}$, $\left| \overline\{A\} \right| \le 2^\{\kappa \}$, then $|X|\le 2^\kappa $; and (b) (Fedeli [2]) If $X$ is a $T_2$-space then $|X|\le 2^\{\operatorname\{aql\}(X)t(X)\psi _c(X)\}$.},
author = {Ramírez-Páramo, Alejandro},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {cardinal functions; cardinal inequalities; Hausdorff space; cardinal functions; cardinal inequalities},
language = {eng},
number = {1},
pages = {131-135},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the cardinality of Hausdorff spaces and Pol-Šapirovskii technique},
url = {http://eudml.org/doc/249542},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Ramírez-Páramo, Alejandro
TI - On the cardinality of Hausdorff spaces and Pol-Šapirovskii technique
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 1
SP - 131
EP - 135
AB - In this paper we make use of the Pol-Šapirovskii technique to prove three cardinal inequalities. The first two results are due to Fedeli [2] and the third theorem of this paper is a common generalization to: (a) (Arhangel’skii [1]) If $X$ is a $T_{1}$ space such that (i) $L(X)t(X)\le \kappa $, (ii) $\psi (X)\le 2^{\kappa }$, and (iii) for all $A \in [X]^{\le 2^{\kappa }}$, $\left| \overline{A} \right| \le 2^{\kappa }$, then $|X|\le 2^\kappa $; and (b) (Fedeli [2]) If $X$ is a $T_2$-space then $|X|\le 2^{\operatorname{aql}(X)t(X)\psi _c(X)}$.
LA - eng
KW - cardinal functions; cardinal inequalities; Hausdorff space; cardinal functions; cardinal inequalities
UR - http://eudml.org/doc/249542
ER -

References

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  1. Arhangel'skii A.V., The structure and classification of topological spaces and cardinal invariants, Russian Math. Surveys (1978), 33-96. (1978) MR0526012
  2. Arhangel'skii A.V., The structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk 33 (1978), 29-84. (1978) MR0526012
  3. Fedeli A., On the cardinality of Hausdorff spaces, Comment. Math. Univ. Carolinae 39.3 (1998), 581-585. (1998) Zbl0962.54001MR1666814
  4. Hodel R., Cardinal functions I, in: K. Kunen, J. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 1-61. Zbl0559.54003MR0776620
  5. Hodel R., A technique for proving inequalities in cardinal functions, Topology Proc. 4 (1979), 115-120. (1979) MR0583694
  6. Juhász I., Cardinal Functions in Topology - Ten Years Later, Mathematisch Centrum, Amsterdam, 1980. MR0576927

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