We introduce the notion of compactifiable classes -these are classes of metrizable compact spaces that can be up to homeomorphic copies "disjointly combined" into one metrizable compact space. This is witnessed by so-called compact composition of the class. Analogously, we consider Polishable classes and Polish compositions. The question of compactifiability or Polishability of a class is related to hyperspaces. Strongly compactifiable and strongly Polishable classes may be characterized by the existence of a corresponding family in the hyperspace of all metrizable compacta. We systematically study the introduced notions -we give several characterizations, consider preservation under various constructions, and raise several questions.Classification: 54D80, 54H05, 54B20, 54E45, 54F15.

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e., a special Peano continuum for which the set containing branchpoints and endpoints has a countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map f of a tame graph G is conjugate to a constant slope map g of a countably affine tame graph. In particular, we show that in the case of a Markov map f that corresponds to recurrent transition matrix, the condition is satisfied for constant slope e htop(f ) , where h top (f ) is the topological entropy of f . Moreover, we show that in our class the topological entropy h top (f ) is achievable through horseshoes of the map f . Classification: Primary 37E25; Secondary 37B40, 37B45.

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We say that two classes of topological spaces are equivalent if each member of one class has a homeomorphic copy in the other class and vice versa. Usually when the Borel complexity of a class of metrizable compacta is considered, the class is realized as the subset of the hyperspace K([0, 1] ω ) containing all homeomorphic copies of members of the given class. We are rather interested in the lowest possible complexity among all equivalent realizations of the given class in the hyperspace.We recall that to every analytic subset of K([0, 1] ω ) there exist an equivalent G δ subset. Then we show that up to the equivalence open subsets of the hyperspace K([0, 1] ω ) correspond to countably many classes of metrizable compacta. Finally we use the structure of open subsets up to equivalence to prove that to every F σ subset of K([0, 1] ω ) there exists an equivalent closed subset.Classification: 54H05, 54B20, 54E45, 54F15.