Generically hereditarily equivalent continua and topological characterization of generic maximal chains of generalized Wa\.zewski dendrites

TL;DR

This paper introduces the concept of generically hereditarily equivalent continua, proves that generalized Ważewski dendrites possess this property, and characterizes the generic maximal chains of their subcontinua.

Contribution

It extends continuum theory by defining generically hereditarily equivalent continua and characterizes generic maximal chains in generalized Ważewski dendrites.

Findings

Generalized Ważewski dendrites are generically hereditarily equivalent.

Existence of a generic orbit under homeomorphism group action.

Topological characterization of generic maximal chains.

Abstract

The notion of hereditarily equivalent continua is classical in continuum theory with only two known nondegenerate examples (arc, and pseudoarc). In this paper we introduce generically hereditarily equivalent continua, i.e. continua which are homeomorphic to comeager many subcontinua. We investigate this notion in the realm of Peano continua and we prove that all the generalized Wa\.zewski dendrites are such. Consequently, we study maximal chains consisting of subcontinua of generalized Wa\.zewski dendrites and we prove that there is always a generic orbit under the homeomorphism group action. As a part of the proof we provide a topological characterization of the generic maximal chain.