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From a Ramsey-type theorem to independence. (English) Zbl 1184.54025

A compact zero-dimensional space \(X\) can be mapped onto the generalized Cantor discontinuum \(D^{\kappa}\) if and only if there exists a binary family \(\tau\) consisting of clopen subsets of \(X\) closed with the respect to complements which fulfils the following conditions
(\(I\)) for all \(S_0, S_1, S_2\in\tau\), if \(S_0\cap S_1=\varnothing\) and \(S_0\cap S_2=\varnothing\) then either \(S_1\cap S_2=\varnothing\) or \(S_1\subset S_2\) or \(S_2\subset S_1\);
(\(T(\kappa)\)) for each subset \(U\in\tau\), \(|\{V\in\tau:V\subset U\}|<\kappa\).
A family \(\tau\) is binary if for each finite subfamily \(\mathcal{M}\subset\tau\) with \(\cap\mathcal{M}=\varnothing\) there exists \(A,B\in\mathcal{M}\) such that \(A\cap B= \varnothing\).

MSC:

54D30 Compactness
05A99 Enumerative combinatorics
05D10 Ramsey theory
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