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In this paper we show that if h : X → X is a mixing homeomorphism on a G-like continuum, then X must be indecomposable and if X is finitely cyclic, then X must be 1 n -indecomposable for some natural number n. Furthermore, we give an example of a mixing homeomorphism on a hereditary decomposable tree-like continuum.

A continuum X has unique cone provided that the following property holds: if Y is a continuum and c o n e ( X ) is homeomorphic to c o n e ( Y ) , then X is homeomorphic to Y. A fan is an arcwise continuum with exactly one ramification point having the property that the intersection of every two subcontinua is connected. This paper is devoted to prove that a fan has unique cone if and only if it is not the cone over a zero-dimensional non-degenerate compact metric space.

For a continuum X, let F n ( X ) be the hyperspace of all nonempty subsets of X with at most n-points. In this paper we show that if X is either a finite graph or a fan (a dendroid with exactly one ramification point) and F n ( X ) is a cone for some n, then X is a cone.

We show that for every upper semicontinuous function f : [ 0 , 1 ] → 2 [ 0 , 1 ] and its graph M = { ( x , y ) ∈ [ 0 , 1 ] 2 : y ∈ f ( x ) } , there is a strong relation between: 1. The generalized inverse limit, lim ← { f , [ 0 , 1 ] } = { ( x 1 , x 2 , … ) ∈ [ 0 , 1 ] N : for each i ∈ N , ( x i + 1 , x i ) ∈ M } = lim ← M and 2. the finite Mahavier product, for each k ∈ N , G k = { ( x 1 , x 2 , … , x k + 1 ) ∈ [ 0 , 1 ] k + 1 : for each i ∈ { 1 , … , k } , ( x i + 1 , x i ) ∈ M } . In particular we show that in the case that lim ← M is one dimensional and arc-wise connected, then the existence of a simple closed curve or the existence of ramification points in M determines the existence of a nonunicoherent continua and ramification points (respectively) in lim ← M and in G k for every k ∈ N . This strong relation allows us to show when lim ← M is a dendrite, this result generalizes the previous results obtained by A. Illanes in [6] and of V. Nall in [13] and [15] for graphs. We obtain the following results: (i) If lim ← M is a dendrite then M and G k are dendrites for every k ∈ N , (ii) if lim ← M is a dendrite with a finite number of ramification points, then M and G k are dendrites with a finite set of ramification points, for every k ∈ N , (iii) if lim ← M is an arc, then M and G k are arcs, for every k ∈ N . All these results allow us to narrow the type of upper semicontinuous functions we can use to obtain a dendrite as a generalized inverse limit.

Given a hereditarily meager ideal I on a countable set X we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology τ I on X such that τ I ∩ I = { ∅ } and, moreover, if I is p + then τ I is selectively separable (SS) and if I is q + , so is τ I . In particular, we obtain regular maximal spaces satisfying all boolean combinations of the properties SS and q + .

Given an arbitrary nondegenerate metric continuum P , we give an uncountable family of topologically distinct metric compactifications of [ 1 , ∞ ) , having P as the remainder.

Given a metric continuum X , we consider the following hyperspaces of X : 2 X , C n ( X ) and F n ( X ) ( n ∈ N ). Let F 1 ( X ) = { { x } : x ∈ X } . A hyperspace K ( X ) of X is said to be rigid provided that for every homeomorphism h : K ( X ) → K ( X ) we have that h ( F 1 ( X ) ) = F 1 ( X ) . In this paper we study under which conditions a continuum X has a rigid hyperspace F n ( X ) . Among others, we consider families of continua such as, dendroids, Peano continua, indecomposable arc continua (all their proper nondegenerate subcontinua are arcs), hereditarily indecomposable continua and smooth fans.

In this paper we show that the space of selections of a smooth fan is a complete separable absolute retract. Then, we show that it is homeomorphic to the Hilbert space l 2 .

We show that under the continuum hypothesis there is a compact zero-dimensional space which admits a base of pairwise homeomorphic clopen subsets but it is not an h -homogeneous space (i.e. not all of its nonempty clopen subsets are homeomorphic), partially answering a question of M.V. Matveev. Under Jensen's ⋄ principle, we can even make the space hereditarily separable and hence, by a result of Matveev, an S -space.

Given a dendroid X, an open selection is an open map s : C ( X ) → X such that s ( A ) ∈ A for every A ∈ C ( X ) . We show that a smooth fan X admits an open selection if and only if X is locally connected.

In this paper we construct an uncountable family of metric compactifications of the ray with the remainder being the pseudo-arc, answering a question posed by Marwan M. Awartani.

Let X be a metric continuum. Let C ( X ) be the hyperespace of subcontinua of X . Given two finite subsets P and Q of X , let U(P,Q)={A∈C(X):P⊂A and A ∩ Q =∅} . In this paper we consider C ( X ) with the topology τ P which have the sets U(P,Q) as a basis. In this paper we show that, for a dendroid X , some topological properties of X are very closely related to the topological structure of ( C ( X ), τ P ) .

In this paper a result of A. Illanes and J.J. Charatonik obtained in [J.J. Charatonik, A. Illanes, Mappings on dendrites, Topology Appl. 144 (2004) 109–132, Corollary 5.14] is extended, by showing that a locally connected continuum X has the nonwandering-eventually-periodic property. (ΩEP-property) iff X is a dendrite that does not contain a homeomorphic copy of the null-comb. Also using “An engine breaking the ΩEP-property” constructed by P. Pyrih et al. in [P. Pyrih, J. Hladký, J. Novak, M. Sterzik, M. Tancer, An engine breaking the ΩEP-property, Topology Appl. 153 (2006) 3621–3626] the results obtained in [J.J. Charatonik, A. Illanes, Mappings on dendrites, Topology Appl. 144 (2004) 109–132; H. Mendez-Lango, On the ΩEP-property, Topology Appl. 154 (2007) 2561–2568] and [P. Pyrih, J. Hladký, J. Novak, M. Sterzik, M. Tancer, An engine breaking the ΩEP-property, Topology Appl. 153 (2006) 3621–3626] are extended, by proving that every nonlocally connected continuum X that contains a nondegenerate arc A and a point p ∈ A such that X is not connected in kleinen at p does not have the ΩEP-property. Answering Question 1 of [P. Pyrih, J. Hladký, J. Novak, M. Sterzik, M. Tancer, An engine breaking the ΩEP-property, Topology Appl. 153 (2006) 3621–3626]. Finally an uncountable family of non-locally connected continua containing arcs with the ΩEP-property is shown.

We show (in ZFC) that if X is a compact homogeneous Hausdorff space then | X | ⩽ 2 t ( X ) , where t ( X ) denotes the tightness of X . It follows that under GCH the character and the tightness of such a space coincide.