  We study direct limits of embedded Cantor sets and embedded \sier curves. We
show that under appropriate conditions on the embeddings, all limits of Cantor
spaces give rise to homeomorphic spaces, called $\omega$-Cantor spaces, and
similarly, all limits of \sier curves give homeomorphic spaces, called to
$\omega$-\sier curves. We then show that the former occur naturally as Morse
boundaries of right-angled Artin groups and fundamental groups of non-geometric
graph manifolds, while the latter occur as Morse boundaries of fundamental
groups of finite-volume, cusped hyperbolic 3-manifolds.
