A theorem on uniform imbedding

Garg, G. L.

doi:10.1007/bf01314687

Cited by 3 publications

(4 citation statements)

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“…It might be possible to prove the metrizability of the adjunction space, without producing any nice explicit metric, by a more direct method, akin to the proof of Corollary 16.2 below. In fact, G. L. Garg gave a (rather complicated) construction of a metrizable uniform space with properties resembling those of the adjunction space [40]; in his Zentralblatt review of Garg's paper, J. R. Isbell claims, without giving any justification, that "the author constructs the pushout of a closed embedding and a surjective morphism in the category of metrizable uniform spaces". We note that due to the nature of Garg's construction, a proof of Isbell's claim would be unlikely to produce any nice explicit metric on the adjunction space.…”

Section: 4mentioning

confidence: 99%

“…Relevance of this argument for the purposes of geometric topology is questionable; one would not be comfortable using this construction as a basis for geometrically substantial results. An alternative, more transparent approach (implicit in the papers of Garg [40] and Nhu [79], [80]; see also [108]) results from replacing embeddings with extremal (or regular) epimorphisms (i.e. embeddings onto closed subsets) in the definitions of A[N]R(MU) and A[N]E(MU ).…”

Section: Uniform Anrsmentioning

confidence: 99%

“…Obviously uniform A[N]Rs include all metrizable A[N]RUs and no other complete spaces. The above proof is much easier than the original one in [40]; a proof that is arguably still easier (as it does not use metrizability of the adjunction space) is presented in [108; Appendix] modelled on an argument in [79].…”

Section: Uniform Anrsmentioning

confidence: 99%

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Metrizable uniform spaces

Melikhov¹

2011

Preprint

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Three themes of general topology: quotient spaces; absolute retracts; and inverse limits -are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include:a uniformly continuous partial map of metric spaces, where A is closed in X, we show that the adjunction space X ∪ f Y with the quotient uniformity (hence also with the topology thereof) is metrizable, by an explicit metric. This yields natural constructions of cone, join and mapping cylinder in the category of metrizable uniform spaces, which we show to coincide with those based on subspace (of a normed linear space); on product (with a cone); and on the isotropy of the l 2 metric.2) We revisit Isbell's theory of uniform ANRs, as refined by Garg and Nhu in the metrizable case. The iterated loop spaces Ω n P of a pointed compact polyhedron P are shown to be uniform ANRs. Four characterizations of uniform ANRs among metrizable uniform spaces X are given: (i) the completion of X is a uniform ANR, and the remainder is uniformly a Z-set in the completion; (ii) X is uniformly locally contractible and satisfies the Hahn approximation property; (iii) X is uniformly ε-homotopy dominated by a uniform ANR for each ε > 0; (iv) X is an inverse limit of uniform ANRs with "nearly splitting" bonding maps. * := C * ∧D X is a uniform cover of X star-refining C and D ′ * := D * ∧C Y is a uniform cover of Y star-refining D. In addition, C ′ * ∩ A = F * = D ′ * ∩ A. Next, let C * * be a uniform cover of X star-refining C ′ * and let D * * be a uniform cover of Y star-refining D ′ * . Let F * * = C * * ∧ D * * , and define C ′ * * and D ′ * * similarly to the above. Then C ′ * * is a uniform cover of X star-refining C ′ * and D ′ * * is a uniform cover of Y star-refining D ′ * ; in addition, C ′

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Section: 4mentioning

confidence: 99%

Section: Uniform Anrsmentioning

confidence: 99%

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Metrizable uniform spaces

Melikhov¹

2011

Preprint

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“…(We can also find the higher splttings and wavefunctions. 24 ) There are three steps in finding the wavefunction itself: (i) find it in the classically allowed region, (ii) find it in the classically forbidden region |m| ≪ S, (iii) match the two parts of the wavefunction using the connection formulas or otherwise. The splitting is then obtainable by a textbook formula.…”

Section: Tunnel Splitting For the Hamiltonian (1)mentioning

confidence: 99%