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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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##### References:
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