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In this article we present a new proof of a sharp Reverse H\"older Inequalityfor $A_\infty$ weights that is valid in the context of spaces of homogeneoustype. Then we derive two applications: a precise open property of Muckenhouptclasses and, as a consequence of this last result, we obtain a simple proof ofa 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 themeasure $\mu$ and the geometric constant $\kappa$ of the quasimetric.

Sharp Reverse H\"older property for A_\infty weights on spaces of homogeneous type