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

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 kind 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 space
• non-cut point
• shore-point

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