  Haslhofer and M\"uller proved a compactness Theorem for four-dimensional
shrinking gradient Ricci solitons, with the only assumption being that the
entropy is uniformly bounded from below. However, the limit in their result
could possibly be an orbifold Ricci shrinker. In this paper we prove a
compactness theorem for noncompact four-dimensional shrinking gradient Ricci
solitons with a topological restriction and a noncollapsing assumption, that
is, we consider Ricci shrinkers that can be embedded in a closed four-manifold
with vanishing second homology group over every field and are strongly
$\kappa$-noncollapsed with respect to a universal $\kappa$. In particular, we
do not need any curvature assumption and the limit is still a smooth nonflat
shrinking gradient Ricci soliton.
