Non-weak cut, non-block, shore, and non-strong center points of $F_n^K(X)$

Year 2025, Early Access, 1 - 9

Roberto Carlos Mondragón Alvarez , Florencio Corona-vázquez , Russell-aaron Quinones-estrella , Javier Sánchez-martínez

https://doi.org/10.15672/hujms.1451354

Abstract

Given a continuum $X$ and a positive integer $n$, $F_{n}(X)$ denotes the hyperspace of all nonempty subsets of $X$ with at most $n$ points endowed with the Hausdorff metric. For $K\in F_{n}(X)$, $F_{n}(K,X)$ denotes the set of all elements of $F_{n}(X)$ containing $K$. We will consider $F_{n}^K(X)$ the quotient space obtained from $F_{n}(X)$ by shrinking $F_{n}(K,X)$ to one point set, endowed with the quotient topology. In this paper, we study the relationship between some types of non-cut points of $F_{n}^{K}(X)$ and the condition of being of the same type of non-cut set over its preimages in $F_{n}(X)$ under the natural quotient map. The non-cut type sets considered here are: non-weak cut, non-block, shore, and non-strong center sets.

Keywords

Continua , hyperspaces , Symmetric products , Quotient spaces

References

• [1] J. G. Anaya and D. Maya, Non-cut ordered arcs of the hyperspace of subcontinua, Topology Appl. 349, 1-10, 108908, 2024.