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A short proof of a theorem of Juhász. (English) Zbl 1229.54004
If \(X\) is a topological space, then \(t(X)\), (respectively, \(\psi(X)\), \( L(X))\) denotes the tightness (respectively, pseudocharacter, Lindelöf number) of \(X\). The cardinal function \(F(X)\) is defined to be the supremum of the lengths of free sequences in \(X\). A theorem originally proved by B. Shapirovskii [Sov. Math., Dokl. 13, 215–219 (1972); translation from Dokl. Akad. Nauk SSSR 202, 779–782 (1972; Zbl 0252.54002)] states that if \(X\) is Hausdorff, then \(|X|\leq 2^{\psi(X)\cdot t(X)\cdot L(X)}\). A generalization by I. Juhász appeared in his book [Cardinal Functions in Topology – ten years later. Mathematical Centre Tracts 123. Amsterdam: Mathematisch Centrum. IV (1980; Zbl 0479.54001)], to the effect that if \(\{X_\alpha:\alpha<\lambda\}\) is a nested family of subsets of \(X\) whose union is \(X\) and \(t(X_\alpha)\cdot\psi(X_\alpha)\cdot L(X_\alpha)\leq\kappa\), then \(|X|\leq 2^\kappa\). The present paper gives a one page proof of this last theorem using elementary submodels and which also shows that \(|X|\leq 2^{\psi(X)\cdot F(X)\cdot L(X)}\).

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