Geodesic
Perfect Images of Zero-Dimensional Separable Metric Spaces
Canadian mathematical bulletin, Tome 25 (1982) no. 1, pp. 41-47

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1982-005-9
Mots-clés : 54 C 10, 54 E 35, perfect continuous surjection, perfect irreducible continuous surjection, separable metric space, zero-dimensional space 
Mill, Jan Van; Woods, R. Grant. Perfect Images of Zero-Dimensional Separable Metric Spaces. Canadian mathematical bulletin, Tome 25 (1982) no. 1, pp. 41-47. doi: 10.4153/CMB-1982-005-9
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[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

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