Metrizable uniform spaces

Abstract

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 ′

Infinite-dimensional uniform polyhedra

“…In particular, the functor respects joins, and mapping cylinders of simplicial maps. Here the join of posets can refer to any of the two wellknown distinct notions, and the join and mapping cylinder of metrizable uniform spaces are defined in [22].…”

“…Theorem 1.6 is arguably the hardest result of the paper. The case of simplicial complexes is somewhat easier (Theorem 4.11), and actually reduces to the case of "cubohedra", which was already treated in [22;Theorem 14.23]. (A cubohedron is a cubical complex that is a subcomplex of the standard cubical lattice in c 0 .)…”

“…Weaker forms of (a) and (b), with cubohedra in place of simplicial complexes, are contained in the author's previous paper [22;Corollary 14.25 and Theorem 18.2]. The present versions, being based on nerves of uniform covers rather than uniform neighborhoods in c 0 , have the advantage of greater flexibility, so they can be straightforwardly adapted, for instance, to equivariant contexts.…”

“…However, if one works in the category of metrizable uniform spaces, then there is no such restriction, as noticed independently by G. L. Garg and N. T. Nhu in the 1970s (see [22; §14]). The resulting theory of (possibly non-complete) uniform ANRs has been developed only recently but turns out to be very flexible [22]. These uniform ANRs are homotopy complete (i.e.…”

“…Specific terms of his program will be reviewed at the end of this introduction. This paper is a common sequel to [21; Chapter 2] and [22] and will in turn be followed by applications to the Hilbert-Smith conjecture.…”