ON SOME STRUCTURAL PROPERTIES OF SPACES OF HOMOGENEOUS TYPE

Abstract

We prove that every space of homogeneous type $(X,\rho,\mu)$ is either an LCH space and $\mu$ is a Radon measure, or $X$ may be identified as a dense subset, with inherited quasi-distance and measure, of another space of homogeneous type which is LCH. We also take an opportunity to present a metamathematical principle which is useful in proving results for general quasi-metric measure spaces by reducing arguments to the case of metric measure spaces.