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Generalized metrizable spaces. (English) Zbl 1314.54001

Hart, K. P. (ed.) et al., Recent progress in general topology III. Based on the presentations at the Prague symposium, Prague, Czech Republic, 2001. Amsterdam: Atlantis Press (ISBN 978-94-6239-023-2/hbk; 978-94-6239-024-9/ebook). 471-505 (2014).
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.
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.
For the entire collection see [Zbl 1282.54001].

MSC:

54-02 Research exposition (monographs, survey articles) pertaining to general topology
54E20 Stratifiable spaces, cosmic spaces, etc.
54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc.
54E30 Moore spaces
54E35 Metric spaces, metrizability
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54H11 Topological groups (topological aspects)
54C35 Function spaces in general topology
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