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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

References

[AU] Alexandroff, P. and Urysohn, P., Über nulldimensionale Punktmemgen, Math. Ann. 98 (1928), 89-106.Google Scholar
[B] Brouwer, L. E. J., On the structure of perfect sets of points, Proc. Akad. Amsterdam 12 (1910), 785-794.Google Scholar
[D] Dugundji, J., Topology, Allyn and Bacon, Boston, 1966.Google Scholar
[E] Engelking, R., Outline of general topology, John Wiley and Sons Inc., New York, 1968.Google Scholar
[GJ] Gillman, L. and Jerison, M., Rings of continuous functions, Van Nostrand, Princeton 1960.Google Scholar
[vM] van Mill, J., Characterization of some zero-dimensional separable metric spaces, pre-print.Google Scholar
[PW] Porter, J. R. and Woods, R. G., Nowhere dense subsets of metric spaces with applications to Stone-Cech compactifications, Canad, J. Math. 24 (1972), 622-630.Google Scholar
[Si] Sierpiński, W., Sur une propriété topologique des ensembles denombrables en soi, Fund. Math. 1 (1920), 44-60.Google Scholar
[Sik] Sikorski, R., Boolean algebras, 2nd Edition (Springer, New York, 1964).Google Scholar
[St] Strauss, D. P., Extremally disconnected spaces, Proc. Amer. Math. Soc. 18 (1967), 305-309.Google Scholar
[T] Tall, F. D., The countable chain condition versus separability?applications of Martin's axiom, Gen. Top. Applic. 4 (1974), 315-339.Google Scholar
[Wa] Walker, R. C., The Stone-Cech compactification, Springer, New York, 1974.Google Scholar
[Wo] Woods, R. G., A survey of absolutes of topological spaces, Top. Structures 2, 1979, (Proceedings of the Amsterdam Symposium) Mathematische Centrum. Tracts, No. 116, 323-362.Google Scholar