  In this work we characterize the sets $E\subset X$ for which there is some
$\alpha>0$ such that the function $d(\cdot,E)^{-\alpha}$ belongs to the
Muckenhoupt class $A_1(X,d,\mu)$, where $(X,d,\mu)$ is a space of homogeneous
type, extending a recent result obtained by Carlos Mudarra in metric spaces
endowed with doubling measures. In particular, generalizations of the notions
of weakly porous sets and doubling of the maximal hole function are given and
it is shown that these concepts have a natural connection with the $A_1$
condition of some negative power of its distance function. The proof presented
here is based on Whitney-type covering lemmas built on balls of a particular
quasi-distance equivalent to the initial quasi-distance $d$ and provided by
Roberto Mac\'ias and Carlos Segovia in "A well-behaved quasi-distance for
spaces of homogeneous type", Trabajos de Matem\'atica 32, Instituto Argentino
de Matem\'atica, 1981, 1-18.
