The projective Fra\"{\i}ss\'{e} limit of the family of all connected finite graphs with confluent epimorphisms

TL;DR

This paper constructs a new continuum as the projective Fraïssé limit of finite connected graphs with confluent epimorphisms, revealing its complex topological properties and embeddings of notable continua.

Contribution

It introduces a novel continuum arising from a projective Fraïssé limit, detailing its indecomposability, topological features, and embeddings of classical continua.

Findings

The continuum is indecomposable but not hereditarily indecomposable.

It is one-dimensional, Kelley, and pointwise self-homeomorphic.

Embeddings of the universal solenoid, pseudo-solenoid, and pseudo-arc are possible.

Abstract

We investigate the projective Fra\"{\i}ss\'e family of finite connected graphs with confluent epimorphisms and the continuum obtained as the topological realization of its projective Fra\"{\i}ss\'e limit. This continuum was unknown before. We prove that it is indecomposable, but not hereditarily indecomposable, one-dimensional, Kelley, pointwise self-homeomorphic, but not homogeneous. It is hereditarily unicoherent and each point is the top of the Cantor fan. Moreover, the universal solenoid, the universal pseudo-solenoid, and the pseudo-arc may be embedded in it.