The hyperspaces $HS(p,X)$

TL;DR

This paper introduces and studies the quotient hyperspace $HS(p,X)$ derived from the hyperspace of subcontinua of a continuum $X$, exploring its properties and relation to $X$ and $C(X)$.

Contribution

It defines the quotient space $HS(p,X)$ and investigates its fundamental properties and connections with the original continuum and its hyperspace.

Findings

Properties of $HS(p,X)$ are characterized.

Relationships between $X$, $C(X)$, and $HS(p,X)$ are established.

Insights into the structure of hyperspaces are provided.

Abstract

Let $X$ be a continuum and let $C(X)$ denote the hyperspace of subcontinua of $X$, endowed with the Hausdorff metric. For $p\in X$, define the hyperspace $C(p,X)=\{A\in C(X):p\in A\}$ as a subspace of $C(X)$. In this paper we introduced the quotient space $HS(p,X)=C(X)/C(p,X)$. We present some general properties of $HS(p,X)$ and we study the relationship between the continuum $X$ and the hyperspaces $C(X)$ and $HS(p,X)$.