  We provide new characterizations of Sobolev spaces that are true under some
mild conditions. We study modified first order Sobolev spaces on metric measure
spaces: $\mathrm{TC}$-Newtonian space, $\hat{\mathrm{TC}}$-Newtonian space, and
Gigli-like space. We prove that if the measure is Borel regular and
$\sigma$-finite, then the modified $\mathrm{TC}$-Newtonian space is equivalent
to the Haj{\l}asz-Sobolev space. Moreover, if additionally the measure is
doubling then all modified spaces are equivalent to the Haj{\l}asz-Sobolev
space.
