Mapping Theorems for Non-Compact Spaces

Dowker, C. H.

doi:10.2307/2371848

Cited by 147 publications

(50 citation statements)

References 4 publications

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“…Pour obtenir un complexe localement de dimension finie, utiliser le lemme 3.3 de [3] dans la démonstration de ce théorème.) Soit (E n , p n , K n ) l'image réciproque de (E, p, B) par g n .…”

Section: Démonstration Du Théorèmeunclassified

Solution d'un probleme de Fadell sur les fibres

Cauty

2001

Colloq. Math.

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Abstract. Let E be the total space of a Hurewicz fiber space whose base and all fibers are ANRs. We prove that if E is metrisable, then it is also an ANR. La démonstration se fera en deuxétapes. Nous prouverons d'abord ce résultat quand B est un complexe simplicial localement de dimension finie avec la topologie métrique, puis nous déduirons le cas général de ce cas particulier. Tous les résultats sur les rétractes absolus de voisinage que nous utiliserons ici se trouvent dans le livre de S. T. Hu [8],à l'exception du suivant dont la démonstration peut se trouver dans [9]. Préliminaires. Nous notons2000 Mathematics Subject Classification: 54C55, 55R05.[151]

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Section: Démonstration Du Théorèmeunclassified

Solution d'un probleme de Fadell sur les fibres

Cauty

2001

Colloq. Math.

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“…These groups denoted by H? (X), have been ignored in favor of those based on arbitrary or locally finite covers, since Dowker showed that H}(X) is not a homotopy invariant of the space X, [7]. However it was subsequently shown in [1] that if k > 1 and G is finitely generated then Hf{ -G) is a homotopy invariant functor on the category of finite dimensional normal spaces.…”

Section: Incompressibility Of Maps and The Homotopy Invariance 15mentioning

confidence: 99%

Incompressibility of maps and the homotopy invariance of Čech cohomology

Calder¹,

Williams²

1981

Pacific J. Math.

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“…Then E Π H and F f] H are freed in if by some subset V with δ Ind F ^ w -1. Applying 2.1 to Jand F, we obtain a subset K oί J satisfying (1) and (2). Then W = K -C -D also satisfies (1) and (2) (the first a fortiori; the second because a set far from K -C -D is the union of a set far from if and a set contained in any preassigned uniform neighborhood of C U D).…”

Section: Lemma Let M Be a Metric Space With Subspaces G And H Then mentioning

confidence: 99%

“…There the dimension function dim is defined, as the covering dimension with respect to the family of all finite normal coverings, and the decidedly imperfect analogy with δd is worked out. Of course dim X = δd aX, where a is the fine uniformity on X. Dowker has given a proof [2] that δd aX = ΔdaX if X is normal (not using this notation); and I pointed out [7] that the same proof 2 shows that δd μX -Δd μX whenever μ is fine or even locally fine.…”

Section: For a Metric Space M If δD M -mentioning

confidence: 99%

On finite-dimensional uniform spaces. II

Isbell¹

1962

Pacific J. Math.

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Introduction. The main subject of this paper is the [inductive* dimension δ Ind μX of uniform spaces μX. This is defined similarly to topological dimension Ind, but instead of separation one uses the notion of a set H, arbitrarily small uniform neighborhoods of which uniformly separate given sets A, B. For finite dimensional metric spaces M (i.e. the large dimension Δd M is finite) 8 Ind coincides with the covering dimensions Ad and δd. For general spaces μX we have 8 Ind μX ^ δd μX. For all known examples (including the examples for Δd Φ δd and, in compact spaces, Ind Φ dim) 8 Ind coincides with δd.The last section of the paper concerns the dimension theory of uniformisable spaces; it organizes alternative definitions and formulates problems, giving limited results on some of the problems. Covering dimension dim has been successfully generalized by Smirnov [17]; here we add to Smirnov's theory a generalization of Aleksandrov's theorem characterizing dim by separating ^-tuples of pairs (A i9 Bi) of disjoint closed sets by closed sets C; with empty intersection. The notion of min dim, mentioned in Part I [7], is formally defined: min dim X is the minimum of Id μX over all compatible uniformities μ. Equivalently, it is the minimum of dim Y over spaces Y containing X. The question when min dim X -dim X, i.e. when X cannot be embedded in a space of lower dimension, is stressed. The Lindelof property implies this, but the question is open for metrizable spaces and more generally for spaces admitting a complete uniformity.It is shown that every completely metrizable space can be homeomorphically embedded as a closed set in a countable product of finitedimensional polyhedra. Combined with results of [9] this means that every completely metrizable space is an inverse limit of polyhedra of the same or lower dimension. The question is still open whether a 1-dimensional completely metrizable space can be an inverse limit of discrete spaces.An announcement of the results on 8 Ind appeared in [8],

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Mapping Theorems for Non-Compact Spaces

Cited by 147 publications

References 4 publications

Solution d'un probleme de Fadell sur les fibres

Solution d'un probleme de Fadell sur les fibres

Incompressibility of maps and the homotopy invariance of Čech cohomology

On finite-dimensional uniform spaces. II

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