  We define a new compactification of outer space $CV_N$ (the \emph{Pacman
compactification}) which is an absolute retract, for which the boundary is a
$Z$-set. The classical compactification $\overline{CV_N}$ made of very small
$F_N$-actions on $\mathbb{R}$-trees, however, fails to be locally $4$-connected
as soon as $N\ge 4$. The Pacman compactification is a blow-up of
$\overline{CV_N}$, obtained by assigning an orientation to every arc with
nontrivial stabilizer in the trees.

Geometric Topology;Group Theory