$R^1$, $R^2$ and $R^3$-continua were defined by S. T. Czuba in 1980, in particular he showed that the existence of any one of these sets in a continuum $X$ implies the noncontractibility of $X$. Also, $R^i$-continua have proved to be useful when studying noncontractibility of hyperspaces. In this talk we recall these concepts and we present some relations between them in continua and hyperspaces.
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Research in general topology is important for the study of dynamical systems. The complexity of dynamical systems suggests the existence of complex topological structures in their base spaces. This lecture will discuss the following two topics: (Part 1) Extended Takens-type reconstruction theorems for one-sided dynamical systems, and (Part 2) The existence of indecomposable continua in chaotic dynamical systems.
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For a continuum, we consider the equivalence relation in which two points are equivalent if they can be joined by an arc. This equivalence relation is analytic in general (i.e. a continuous image of a Polish space). Recently, Debs and Saint Raymond proved that, for planar continua, this equivalence relation is always Borel measurable. We show that for every planar continuum, the arc-connection relation is in fact Borel reducible to the Vitali equivalence relation, where two real numbers are equivalent if their difference is rational. Moreover, the Knaster continuum is an example where this complexity is attained. This is joint work with Michal Hevessy and Yusuf Uyar. We also investigate several related questions concerning continuum-wise connectivity.
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