On $s$-sets in spaces of homogeneous type
Let $(X,d,\mu )$ be a space of homogeneous type. We study the relationship between two types of $s$-sets: relative to a distance and relative to a measure. We find a condition on a closed subset $F$ of $X$ under which $F$ is an $s$-set relative to the measure $\mu $ if and only if $F$ is an $s$-set relative to $\delta $. Here $\delta $ denotes the quasi-distance defined by Macías and Segovia such that $(X,\delta ,\mu )$ is a normal space. In order to prove this result, we prove a covering type lemma and a type of Hausdorff measure based criterion for a given set to be an $s$-set relative to $\mu $.