Abstract. In a nondegenerate continuum we study the set of noncut points. We show that it can be stratified by inclusion into six natural subsets (containing also non-block and shore points). Among other results we show that every nondegenerate continuum contains at least two nonblock points. Our investigation is further focused on both the classes of arc-like and circle-like continua.

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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