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On the cardinality of Hausdorff spaces and Pol-Šapirovskii technique. (English) Zbl 1121.54013
Summary: In this paper we make use of the Pol-Šapirovskii technique to prove three cardinal inequalities. The first two results are due to A. Fedeli [Commentat. Math. Univ. Carol. 39, 581–585 (1998; Zbl 0962.54001)] and the third theorem of this paper is a common generalization to:
(a) 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) If \(X\) is a \(T_2\)-space then \(| X| \leq 2^{\text{aql}(X)t(X) \psi _c(X)}\).

MSC:
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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