  Let $(\mathcal{X},d,\mu)$ be a metric measure space satisfying the so-called
upper doubling condition and the geometrically doubling condition. Let $T$ be a
Calder\'{o}n-Zygmund operator with kernel satisfying only the size condition
and some H\"ormander-type condition, and $b\in\rm{\widetilde{RBMO}(\mu)}$ (the
regularized BMO space with the discrete coefficient). In this paper, the
authors establish the boundedness of the commutator $T_b:=bT-Tb$ generated by
$T$ and $b$ from the atomic Hardy space $\widetilde{H}^1(\mu)$ with the
discrete coefficient into the weak Lebesgue space $L^{1,\,\infty}(\mu)$. The
boundedness of the commutator generated by the generalized fractional integral
$T_\alpha\,(\alpha\in(0,1))$ and the $\rm{\widetilde{RBMO}(\mu)}$ function from
$\widetilde{H}^1(\mu)$ into $L^{1/{(1-\alpha)},\,\fz}(\mu)$ is also presented.
Moreover, by an interpolation theorem for sublinear operators, the authors show
that the commutator $T_b$ is bounded on $L^p(\mu)$ for all $p\in(1,\infty)$.
