# Homogeneity degree of fans

## TL;DR

This paper studies the homogeneity degree of fans, a class of topological spaces, classifies smooth fans with degree 3, and explores properties of non-smooth fans with degree 4.

## Contribution

It provides a classification of smooth fans with homogeneity degree 3 and advances understanding of non-smooth fans with degree 4.

## Key findings

- Classified all smooth fans of homogeneity degree 3.
- Proved results on non-smooth fans with degree 4.
- Initiated study of small homogeneity degree dendroids.

## Abstract

The homogeneity degree of a topological space $X$ is the number of orbits of the action of the homeomorphism group of $X$ on $X$. We initiate a study of dendroids of small homogeneity degree, beginning with fans. We classify all smooth fans of homogeneity degree $3$, and discuss non-smooth fans and prove some results on degree $4$.