Nonseparable analytic metric spaces and quotient maps

✍ Scribed by R.W. Hansell

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 657 KB
• Volume: 85
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(97)00146-6

✦ Synopsis

E. Michael and A.H. Stone have shown that any separable analytic metric space is an almostopen continuous image of the space ~2'. Hence the separable analytic metric spaces are precisely the quotients of the space of irrational numbers. Here it is shown that the nonseparable analytic metric spaces are precisely the continuous almost-open s-images of the closed subspaces of the Baire spaces K"', where by an s-image we mean all fibres of the map are separable. A similar result is obtained in which the domain is all of K~. Further, it is shown that any metric continuous quotient s-image of an analytic metric space is again analytic, and examples are given to show why the various assumptions are necessary. 0 1998 Elsevier Science B.V.