In this paper, we establish the equivalence between the Haj{\l}asz-Sobolev
spaces or classical Triebel-Lizorkin spaces and a class of grand
Triebel-Lizorkin spaces on Euclidean spaces and also on metric spaces that are
both doubling and reverse doubling. In particular, when $p\in(n/(n+1),\fz)$, we
give a new characterization of the Haj{\l}asz-Sobolev spaces $\dot M^{1,
p}(\rn)$ via a grand Littlewood-Paley function.