Quotients of the space of irrationals
1969; Mathematical Sciences Publishers; Volume: 28; Issue: 3 Linguagem: Inglês
10.2140/pjm.1969.28.629
ISSN1945-5844
Autores Tópico(s)Advanced Topics in Algebra
ResumoIt is proved that every metric space which is a continuous image of the irrationals is also a quotient of the irrationals.In this paper we are concerned with the class j^ of all those metric spaces which are continuous images of complete separable metric spaces.The members of Szf are generally called "(absolutely) analytic sets" or "A-sets" [9] or "Souslin spaces" [5], and are known to be precisely those metric spaces which are either empty or are continuous images of the space P of irrational numbers 1 .Suppose, then, that Y e Jzf and Y is nonempty.There exists a continuous surjection f:P->Y; how "nice" can / be taken to be?In general, / cannot be one-toone (or Y would have to be absolutely Borel; see [9 p. 487]); nor can / be open or closed (as Y would then be an absolute G δ ; see 3.4 and 3.5 below).However, we shall see that / can always be chosen to be a quotient map.More precisely, we prove the following theorem.THEOREM 1.1.Every metrίzable space Y which is a continuous image of P is also a quotient of P (under a different map, in general).
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