the Lelek fan and for the [Fraïssé ]{}fence).

-   Both are universal in the respective classes with respect to embeddings that preserve endpoints (see [@Dijkstra2010] for the Lelek fan and for the [Fraïssé ]{}fence).

-   For both, the set of endpoints is dense (see for the [Fraïssé ]{}fence). In fact, the Lelek fan is defined as the unique smooth fan with a dense set of endpoints; the [Fraïssé ]{}fence too has a characterization in terms of denseness of endpoints (see ).

-   The set of endpoints of the Lelek fan is homeomorphic to the complete Erdős space ([@MR1391294]), a homogeneous, almost zero-dimensional, $1$-dimensional space; the complete Erdős space is cohesive, that is, every point has a neighborhood which does not contain any nonempty clopen subset. Among the subspaces of the set of endpoints of the [Fraïssé ]{}fence there is a homogenous, almost zero-dimensional, $1$-dimensional space $ \mathfrak M $ which is not cohesive ((iv)).

A space with the properties mentioned for $ \mathfrak M $ was constructed in [@Dijkstra2006] as a counterexample to a question by Dijkstra and van Mill. This raises the question of whether the two examples are homeomorphic and whether they can be regarded as a non-cohesive 