General Topology III

The problem of metrization of topological spaces has had an enormous influence on the development of general topology. Singling out the basic topo- logical components of metrizability has determined the main reference points in the construction of the classification of topological spaces. These are (pri- marily) paracompactness, collectionwise normality, monotonic normality and perfect normality, the concepts of a stratifiable space, Moore space and u- space, point-countable base, and uniform base. The method of covers has taken up a leading role in this classification. Of paramount significance in the applications of this method have been the properties of covers relating to the character of their elements (open covers, closed covers), the mutual dispo- sition of these elements (star finite, point finite, locally finite covers, etc. ), as well as the relations of refinement between covers (simple refinement, refine- ment with closure, combinatorial refinement, star and strong star refinement). On this basis a hierarchy of properties of paracompactness type has been sin- gled out, together with the classes of spaces corresponding to them, the most important of which is the class of paracompacta. The behaviour of families of covers with respect to the topology of a space has important significance. Here, first and foremost, is the notion of a refining family of covers, a development which appears in several modifications and, together with the notion of paracompactness, plays a key role in metrization problems.