The Lelek fan and the Poulsen simplex as Fra\"iss\'e limits

TL;DR

This paper characterizes the Lelek fan and Poulsen simplex as Fraïssé limits, establishing their uniqueness and universality, and demonstrating their high degree of symmetry through automorphisms.

Contribution

It introduces a Fraïssé-theoretic framework for the Lelek fan and Poulsen simplex, providing new proofs of their key properties and enhancing understanding of their automorphism groups.

Findings

Lelek fan and Poulsen simplex are Fraïssé limits.

Established their uniqueness, universality, and almost homogeneity.

Proved high symmetry of the Lelek fan with automorphisms mapping dense end-point sets.

Abstract

We describe the Lelek fan, a smooth fan whose set of end-points is dense, and the Poulsen simplex, a Choquet simplex whose set of extreme points is dense, as Fra\"{i}ss\'e limits in certain natural categories of embeddings and projections. As an application we give a short proof of their uniqueness, universality, and almost homogeneity. We further show that for every two countable dense subsets of end-points of the Lelek fan there exists an auto-homeomorphism of the fan mapping one set onto the other. This improves a result of Kawamura, Oversteegen, and Tymchatyn from 1996.