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 $n\ge 2$.