Generalization of \(G^*_\delta\)-diagonals and \(w\Delta\)-spaces
The concept of \(G^*_\delta\)-diagonal spaces as a generalization of metric spaces is the subject of additional generalization to the concept of quasi-\(G^*_\delta\)-diagonal and quasi-\(w\Delta\)-spaces. The quasi-\(w\Delta\)-space concept is in fact the generalization of \(w\Delta\)-space concept. It is shown that a quasi-Moore space can be characterized in terms of these concepts introduced by the author. As a consequence the following metrization theorems are proved: 1. Every paracompact \(w\Delta\)-space with quasi-\(G^*_\delta\)-diagonal is metrizable. 2. Every collectionwise normal \(\delta\) quasi-\(w\Delta\)-space is metrizable. The main result is the following extension of Hodel's theorem: A space is a quasi-Moore space if and only if it is regular quasi-\(w\Delta\)-space with quasi-\(G^*_\delta\)-diagonal. A space is developable if and only if it is a perfect, quasi-\(w\Delta\)-space with a quasi-\(G^*_\delta\)-diagonal. The complete clarification of relationships between the classes of developable spaces, \(w\Delta\)-spaces, quasi-developable spaces, perfect spaces, quasi-\(w\Delta\)-spaces and quasi-\(G^*_\delta\)-diagonal spaces are also given.
scientific article; zbMATH DE number 1261216