Perfect Images of Zero-dimensional Separable Metric Spaces
نویسندگان
چکیده
Let Q denote the rationals, P the irrationals, C the Cantor set and L the space C {p} (where p e C). Let / : X —> Y be a perfect continuous surjection. We show: (1) If X G { Q , P, QxP} , or if / is irreducible and Xe{C, L}, then Y is homeomorphic to X if Y is zero-dimensional. (2) If X G {P, C, L} and / is irreducible, then there is a dense subset S of Y such that / | /*~[S] is a homeomorphism onto S. However, if Z is any .«--compact nowhere locally compact metric space then there is a perfect irreducible continuous surjection from Q x C onto Z such that each fibre of the map is homeomorphic to C. §
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