Let $S$ be a dominating subset of $[\mathbb{N}]^{\infty}$. Then $S=\{s_{\alpha} : \alpha<\mathfrak{d}\}$ is called a $\mathfrak{d}$-scale if for every $\alpha<\beta<\mathfrak{d}$, $s_{\beta}\nleq^{*} s_{\alpha}$. In the paper of Tsaban, there is a lemma: every $\mathfrak{d}$-scale is $\mathfrak{d}$-concentrated on $[\mathbb{N}]^{<\infty}$. How can we prove this lemma? It is obvious that every scale is $\mathfrak{d}$-concentrated on $[\mathbb{N}]^{<\infty}$, but every $\mathfrak{d}$-scale?
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1$\begingroup$ don't you want to tell us which paper you are referring to? $\endgroup$– Carlo BeenakkerCommented Apr 9, 2021 at 7:45
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$\begingroup$ I forgot it sir. I am sorry, the paper is MENGER’S AND HUREWICZ’S PROBLEMS: SOLUTIONS FROM “THE BOOK” AND REFINEMENTS @CarloBeenakker $\endgroup$– Dans0804Commented Apr 9, 2021 at 7:48
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There is one proof in Theorem 1.7 of the paper you cited. But a simpler argument is provided in Products of Menger spaces: A combinatorial approach (with P. Szewczak), Annals of Pure and Applied Logic 168 (2017). See 2.1-2.3 in this paper.
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$\begingroup$ Thank you so much sir, I got it. @BoazTsaban $\endgroup$– Dans0804Commented Apr 9, 2021 at 10:34