Muckenhoupt weights with singularities on closed lower dimensional sets in spaces of homogeneous type

We give sufficient conditions on a real number β and on a closed set F in a general space of homogeneous type (X,d,μ) in such a way that μ(B(x,d(x,F)))β becomes a Muckenhoupt weight. In order to prove our result, we modify the underlying space so that it becomes 1-Ahlfors regular.