  In this article we present a new proof of a sharp Reverse H\"older Inequality
for $A_\infty$ weights that is valid in the context of spaces of homogeneous
type. Then we derive two applications: a precise open property of Muckenhoupt
classes and, as a consequence of this last result, we obtain a simple proof of
a sharp weighted bound for the Hardy-Littlewood maximal function involving
$A_\infty$ constants: |M|_{L^p(w)} \leq c (\frac{1}{p-1}
[w]_{A_p}[\sigma]_{A_\infty})^{1/p}, where $1<p<\infty$,
$\sigma=w^{\frac{1}{1-p}}$ and $c$ depends only on the doubling constant of the
measure $\mu$ and the geometric constant $\kappa$ of the quasimetric.
