N}}$ converging to $x$, the sequence $([t, x_{n}])_{n \in {\mathbb N}}$ of arcs connecting $t$ to $x_{n}$ converges to $[t, x]$. Smooth fans where introduced in [@MR0227944] and have been extensively studied in continuum theory. A point $x$ in a topological space $X$ is an *endpoint* if whenever $x$ belongs to an arc $[a, b] \subseteq X$, then $x = a$ or $x = b$ (note that under this definition points whose connected component is a singleton are endpoints). A *Lelek fan* is a smooth fan with a dense set of endpoints. Such a fan was first constructed in [@MR133806] and was later proven to be unique up to homeomorphism in [@MR991691] and [@MR1002079]. In a series of papers ([@Bartos2015; @Bartos2017; @MR3939568]) Bartošová and Kwiatkowska have studied the Lelek fan and the dynamics of its homeomorphism group by realizing it as a quotient of a projective Fraïssé limit of a particular class of ordered structures.

Besides the fact that both can be obtained as quotients of projective [Fraïssé ]{}limits of some class of ordered structures, the [Fraïssé ]{}fence and the Lelek fan share several other features:

-   Both are as homogeneous as possible, namely they are $\nicefrac{1}{3}$-homogeneous (see [@MR3712972] for 