October 8, 2025Open Access

Contractibility of the hyperspace of meager subcontinua, two examples

AIAlejandro IllanesUniversidad Nacional Autónoma de México DADiego Alexander Ramírez AngaritaUniversidad Nacional Autónoma de México

Key Points

No implication exists between the conditions of contractibility for the continuum X and its hyperspace M(X).
The study constructs two specific dendroids to illustrate the independent nature of these contractibility conditions.
For arc-smooth continua, it is shown that their corresponding hyperspace M(X) retains contractibility.
The findings build on inquiries posed by Norberto Ordoñez regarding continuity in metric spaces.

Abstract

Given a metric continuum X, the hyperspace of meager subcontinua of X, denoted by M(X), is the collection of all subcontinua of X with empty interior. Answering questions by Norberto Ordo˜nez, in this paper we construct two dendroids showing that there is no implication between the conditions: (a) X is contractible and (b) M(X) is contractible. We also prove that if X is an arc-smooth continuum, then M(X) is contractible.

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