Given a continuum $X$ (compact metrizable connected $X$) let $K(X)$ denote the hyperspace of nonempty compact subspaces of $X$ with the Vietoris topology and let $C(X)$ denote the (closed) subspace of $K(X)$ consisting of connected sets.

It is well known (and has been discussed on MO before) that if there is a continuous choice function $K(X)\to X$ then $X\simeq[0,1]$, indeed any other continua doesn’t even have a continuous choice function for the subspace of $K(X)$ of unordered pairs.

The situation is different when looking at $C(X)$: for example the “star” $X$ obtained by joining $n$ arcs at their endpoint has a continuous selection for $C(X)$ by looking at the tree partial order induced by choosing the common point as root and then mapping $C\in C(X)$ to its $\min$ with respect to this partial order. In a similar way we get a continuous selection for $C(X)$ whenever $X$ is a dendrite.

It is also easy to see that $S^1$ doesn’t have a continuous selection for $C(S^1)$ by using a combination of Borsuk-Ulam, $C(S^1)\simeq D^2$ and $\partial C(S^1)\cong F_1(S^1)$, where $D^2$ is the unit disk in $\Bbb R^2$ and $F_1(X)$ is the space of singletons of $X$.

This suggests the following: suppose $X$ is a continuum with a continuous selection $C(X)\to X$. Must $X$ be a subcontinuum of a dendrite? If not is there a characterization of the continua $X$ with a continuous selection $C(X)\to X$?

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