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The pseudo-arc is the unique hereditarily indecomposable chainable continuum.

The Lelek fan is the unique compact, connected subset of the Cantor fan (the cone over the Cantor set) with a dense endpoint set. The Lelek fan is a decomposable continuum and thus does not embed into the pseudo-arc.

Question. Does the set $E(L)$ of endpoints of the Lelek fan embed into the pseudo-arc?

This is equivalent to asking whether the pseudo-arc contains Erdos space $\mathfrak E$, as each of $E(L)$ and $\mathfrak E$ contains the other.

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    $\begingroup$ The pseudoarc $P$ contains a topological copy of $C\times P$, where $C$ is the Cantor set: (a) $P$ is not Suslinean, (b) any nondegenerate subcontinuum of $P$ is homeomorphic to $P$, (c) by a result of van Douwen 1993 $P$ contains a copy of $C\times P$. Hence your question is equivalent to: Does $C\times P$ contains the Erdös space? I suspect that this is true, since a similar process as in the construction of Erdös space inside $C\times [0,1]$ can be followed. Of course, the details need to be checked (e.g. using a topological characterization of Erdös space). $\endgroup$ May 12, 2023 at 6:36
  • $\begingroup$ Perhaps a Cantor-set-valued function can be defined from $C$ into $P$, using the arclength function on the Lelek fan, so that the graph (as a subspace of $C\times P$) is homeomorphic to $C\times E(L)$, which is homeomorphic to $E(L)$. $\endgroup$ May 12, 2023 at 19:38

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