Let A denote the Alexandroff-Urysohn double arrow space. We prove the following results: (a) A× ω 2 is not countable dense homogeneous; (b) ω A is not countable dense homogeneous; (c) A has exactly c types of countable dense subsets. These results answer questions by Arhangel'skiȋ, Hrušák and van Mill.

We show that all sufficiently nice λ-sets are countable dense homogeneous (CDH). From this fact we conclude that for every uncountable cardinal κ ≤ b there is a countable dense homogeneous metric space of size κ. Moreover, the existence of a meager in itself countable dense homogeneous metric space of size κ is equivalent to the existence of a λ-set of size κ. On the other hand, it is consistent with the continuum arbitrarily large that every CDH metric space has size either ω 1 or size c. An example of a Baire CDH metric space which is not completely metrizable is presented. Finally, answering a question of Arhangel'skii and van Mill we show that that there is a compact non-metrizable CDH space in ZFC.

Abstract. We study relations between transitivity, mixing and periodic points on dendrites. We prove that when there is a point with dense orbit which is not an endpoint, then periodic points are dense and there is a terminal periodic decomposition (we provide an example of a dynamical system on a dendrite with dense endpoints satisfying this assumption). We also show that it may happen that all periodic points except one (and points with dense orbit) are contained in the (dense) set of endpoints. It may also happen that dynamical system is transitive but there is a unique periodic point, which in fact is the unique fixed point. We also prove that on almost meshed-continua (a class of continua containing topological graphs and dendrites with closed or countable set of endpoints), periodic points are dense if and only if they are dense for the map induced on the hyperspace of all nonempty compact subsets.

We prove that if F is a non-meager P -filter, then both F and ω F are countable dense homogeneous spaces.

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