Daria Michlik: Symmetric products as cones

Tuesday, January 8, 2019, 17:15
Wrocław University of Technology, 215 D-1

Speaker: Daria Michlik (Cardinal Stefan Wyszynski University in Warsaw)

Title: Symmetric products as cones

Abstract:
(join work with Alejandro Illanes and Veronica Martinez-de-la-Vega)
For a continuum $X$, let $F_n(X)$ be the hyperspace of all nonempty subsets of $X$ with at most $n$-points. The space $F_n(X)$ is called the n’th-symmetric product.
In
A. Illanes, V. Martinez-de-la-Vega, Symmetric products as cones, Topology Appl. 228 (2017), 36–46
it was proved that if $X$ is a cone, then its hyperspace $F_n(X)$ is also a cone.
During my talk I will discuss the converse problem. I will prove that if $X$ is a locally connected curve, then the following conditions are equivalent:
1. $X$ is a cone,
2. $F_n(X)$ is a cone for some $n\ge 2$,
3. $F_n(X)$ is a cone for each $n\ge 2$.

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