We discuss and illustrate a key ingredient to a recent positive solution [Brazas, Conner, F, Kent] to the following problem posed by Jerzy Dydak in 2011.

If the continuous surjection $\Pi :X\rightarrow D^{2}$ has unique based path lifting, must $\Pi$ be a homeomorphism, provided $X$ is a connected, locally path connected metric space and $D^{2}$ is the unit disk?

The answer is “yes”, but ruling out the possibility of a counterexample is nontrivial, and ultimately reduces to the question of whether $X$ could be a certain topological R-tree comprised of all $p$ based irreducible paths in $% D^{2}$.

The irreducible paths $\alpha$ in $D^{2}$ are those such that every nonconstant subloop of $\alpha$ fails to lift to some loop in some dendrite. For example piecewise linear, and more geneally, piecewise irreducible paths in $D^{2}$ lift uniquely to $X,$ up to basepoint.

The challenge is to exhibit a path in $D^{2}$ which does not lift uniquely to $X.$ Illustrating a method to do this is the main goal of the talk, and the tactic is as follows.

Every dendrite is a quotient of a topological disk so that each point preimage intersects the boundary of the disk.

However, mating two respective dendrite partitions of two unit half disks, reveals that the join of the quotients need not be a dendrite.