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Perfect Images of Zero-Dimensional Separable Metric Spaces

Published online by Cambridge University Press:  20 November 2018

Jan Van Mill  and
R. Grant Woods
Jan Van Mill
Affiliation:
Louisiana State University, Baton Rouge, Louisiana
R. Grant Woods
Affiliation:
University of Manitoba, Winnipeg, Canada
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Abstract

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Let Q denote the rationals, P the irrationals, C the Cantor set and L the space C − {p} (where p ∈ C). Let f : X → Y be a perfect continuous surjection. We show: (1) If X ∈ {Q, P, Q × P}, or if f is irreducible and X ∈ {C, L}, then Y is homeomorphic to X if Y is zero-dimensional. (2) If X ∈ {P, C, L} and f is irreducible, then there is a dense subset S of Y such that f|f ← [S] is a homeomorphism onto S. However, if Z is any σ-compact nowhere locally compact metric space then there is a perfect irreducible continuous surjection from Q × C onto Z such that each fibre of the map is homeomorphic to C.

Keywords

54 C 1054 E 35perfect continuous surjectionperfect irreducible continuous surjectionseparable metric spacezero-dimensional space
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 1 , 01 March 1982 , pp. 41 - 47
Copyright
Copyright © Canadian Mathematical Society 1982

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