# Article

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Keywords:
cardinal functions; cardinal inequalities; Hausdorff space
Summary:
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) $L(X)t(X)\leq\kappa$, (ii) $\psi(X)\leq 2^{\kappa}$, and (iii) for all $A \in [X]^{\leq 2^{\kappa}}$, $\left| \overline{A} \right| \leq 2^{\kappa}$, then $|X|\leq 2^\kappa$; and (b) (Fedeli [2]) If $X$ is a $T_2$-space then $|X|\leq 2^{\operatorname{aql}(X)t(X)\psi_c(X)}$.
References:
[1] Arhangel'skii A.V.: The structure and classification of topological spaces and cardinal invariants. Russian Math. Surveys (1978), 33-96. MR 0526012
[1] Arhangel'skii A.V.: The structure and classification of topological spaces and cardinal invariants. Uspekhi Mat. Nauk 33 (1978), 29-84. MR 0526012
[2] Fedeli A.: On the cardinality of Hausdorff spaces. Comment. Math. Univ. Carolinae 39.3 (1998), 581-585. MR 1666814 | Zbl 0962.54001
[3] Hodel R.: Cardinal functions I. in: K. Kunen, J. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, pp. 1-61. MR 0776620 | Zbl 0559.54003
[4] Hodel R.: A technique for proving inequalities in cardinal functions. Topology Proc. 4 (1979), 115-120. MR 0583694
[5] Juhász I.: Cardinal Functions in Topology - Ten Years Later. Mathematisch Centrum, Amsterdam, 1980. MR 0576927

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