Generalized metrizable spaces (Q2874864)

This survey article on generalized metrizable spaces is the fourth in a series that was started by the author in [Handbook of set-theoretic topology, 423--501 (1984; Zbl 0555.54015)]; Husek, Miroslav (ed.) et al., Recent progress in general topology. Papers from the Prague Toposym 1991, held in Prague, Czechoslovakia, 1991. Amsterdam: North-Holland. 239--274 (1992; Zbl 0794.54034), and Husek, Miroslav (ed.) et al., Recent progress in general topology II. Based on the Prague topological symposium, Prague, Czech Republic, 2001. Amsterdam: Elsevier. 201--225 (2002; Zbl 1029.54036)]. It covers results obtained from approximately 2001 till 2013. Since the author regards ``a class of spaces to be a class of generalized metrizable spaces if every metrizable space is in the class, and if the defining property of the class gives its members enough structure to make a reasonably rich and interesting theory,'' many classes could have been considered. However, this time he concentrates on the following themes.NEWLINENEWLINE 1. Around Regular \(G_\delta\)-diagonals; -- 2. Small Diagonal; -- 3. Continuously Urysohn, 2-Maltsev, and (P); -- 4. Gruenhage Spaces and Property (*); -- 5. Stratifiable vs. \(M_1\); -- 6. Stratifiability of Function Spaces; -- 7. Local Versions of \(M_i\)-spaces; -- 8. Quarter-Stratifiable Spaces; -- 9. Compact \(G_\delta\)-sets and \(c\)-semistratifiable spaces; -- 10. Cosmic Spaces; -- 11. \(k^*\)-metrizable Spaces; -- 12. D-spaces; -- 13. Monotone Normality and Resolvability; -- 14. Monotonically Monolithic and Collins-Roscoe Condition (G); -- 15. Monotonically Compact and Monotonically Lindelöf; -- 16. Monotonically Countably Paracompact (MCP); -- 17. \(\beta\)- and strong \(\beta\)-spaces; -- 18. Noetherian Type; -- 19. Base Paracompact; -- 20. Sharp Base; -- 21. \(\text{dis}(X)\) and \(m(X)\); -- 22. Generalized Metrizable Spaces and Topological Algebra; -- 23. Domain Representability.NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].