Ri-continua and hyperspaces

AbstractIt is proved that if a continuum X contains an Ri-continuum for some iϵ{1,2,3}, then the hyperspaces 2x and C(X) contain Ri-continua, therefore they are not contractible. Moreover, 2x has no confluent Whitney map. Some examples concerning this subject are given and som

Projective Fra\"{\i}ss\'{e} limits of trees

We continue study of projective Fra\"{\i}ss\'e limit developed by Irvin, Panagiotopoulos and Solecki. We modify the ideas of monotone, confluent, and light mappings from continuum theory as well as several properties of continua so as to apply to topological graphs. As the topological realizations of the projective Fra\"{\i}ss\'e limits we obtain the dendrite $D_3$ as well as quite new, interesting continua for which we do not yet have topological characterizations

The projective Fra\"{\i}ss\'{e} limit of the class of all connected finite graphs with confluent epimorphisms

We show that the class of finite connected graphs with confluent epimorphism is a projective Fra\"{\i}ss\'e class and we investigate the continuum obtained as the topological realization of the projective Fra\"{\i}ss\'e limit. This continuum was unknown before. We show that it is indecomposable, but not hereditarily indecomposable, one-dimensional, Kelley, pointwise self-homeomorphic, but not homogeneous. It is hereditarily unicoherent and each point is the top of the Cantor fan. Moreover, the universal solenoid, the universal pseudo-solenoid, and the pseudo-arc may be embedded in it

Atomoicity of mappings

A mapping f:X→Y between continua X and Y is said to be atomic at a subcontinuumK of the domain X provided that f(K) is nondegenerate and K=f−1(f(K)). The set of subcontinua at which a given mapping is atomic, considered as a subspace of the hyperspace of all subcontinua of X, is studied. The introduced concept is applied to get new characterizations of atomic and monotone mappings. Some related questions are asked

Smoothness and the property of Kelley

summary:Interrelations between smoothness of a continuum at a point, pointwise smoothness, the property of Kelley at a point and local connectedness are studied in the paper

Monotone homogeneity of dendrites

summary:Sufficient as well as necessary conditions are studied for a dendrite or a dendroid to be homogeneous with respect to monotone mappings. The obtained results extend ones due to H. Kato and the first named author. A number of open problems are asked

On Mazurkiewicz sets

summary:A Mazurkiewicz set $M$ is a subset of a plane with the property that each straight line intersects $M$ in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set

A continuum XX which has no confluent Whitney map for 2\sp{X}

An example is shown of a continuum X which has no confluent Whitney map for 2X. This answers two problems asked by Nadler [N]

Induced near-homeomorphisms

summary:We construct examples of mappings $f$ and $g$ between locally connected continua such that $2^f$ and $C(f)$ are near-homeomorphisms while $f$ is not, and $2^g$ is a near-homeomorphism, while $g$ and $C(g)$ are not. Similar examples for refinable mappings are constructed