The paper is focused on the so called localization problem which can be characterised as a determination of certain topological features within a space. While the topology, and especially the homology gives the information about the existence of the topological features and attributes, the information about their location cannot be directly known. The described method -- localized homology -- assumes that the given space is equipped with a cover without any restrictions on the geometry or the topology of this cover. First, the whole cover is considered to find an initial localization and, second, the recursive refinement is used to localize the geometry contained within the cover. The theoretical foundations for localization of arbitrary-dimensional attributes in arbitrary spaces are given in this paper. The localization problem is a very important topic in many areas -- e.g. in computer graphics, sensor networks, robotic, etc.