## 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

### Keywords:

Cantor cube; binary family
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