A compact Hausdorff space all of whose infinite closed subsets are $n$-dimensional

It is proved that for every $n$ there exists an $n$-dimensional bicompactum (= compact Hausdorff space) with first axiom of countability, such that every closed subset $F$ has either $\dim F\leqslant0$ or $\dim_GF=n$, where $G$ is an arbitrary nonzero Abelian group. The main result is the construction, for every $n\geqslant1$, assuming the continuum hypothesis, of an $n$-dimensional bicompactum of which every closed subset is either finite or $n$-dimensional.
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