Perfect Images of Zero-Dimensional Separable Metric Spaces
Canadian mathematical bulletin, Tome 25 (1982) no. 1, pp. 41-47
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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.
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
@article{10_4153_CMB_1982_005_9,
author = {Mill, Jan Van and Woods, R. Grant},
title = {Perfect {Images} of {Zero-Dimensional} {Separable} {Metric} {Spaces}},
journal = {Canadian mathematical bulletin},
pages = {41--47},
year = {1982},
volume = {25},
number = {1},
doi = {10.4153/CMB-1982-005-9},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1982-005-9/}
}
TY - JOUR AU - Mill, Jan Van AU - Woods, R. Grant TI - Perfect Images of Zero-Dimensional Separable Metric Spaces JO - Canadian mathematical bulletin PY - 1982 SP - 41 EP - 47 VL - 25 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1982-005-9/ DO - 10.4153/CMB-1982-005-9 ID - 10_4153_CMB_1982_005_9 ER -
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