On s-set in space of homogeneous type

Abstract

Let (X, d, μ) 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 μ if and only if F is an s-set relative to δ. Here δ denotes the quasi-distance defined by Macías and Segovia such that (X, δ, μ) 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 μ.