Functorial uniformization of topological spaces

Abstract

We show that there exists at least a proper class of functorial approach structures, i.e., right inverses to the forgetful functor T : AP → Top (where AP denotes the topological construct of approach spaces and contractions as introduced by R. Lowen). There is however a great difference in nature of these functorial approach structures when compared to the quasi-uniform paradigm which has been extensively studied by the first author: whereas it is well-known from [2] that a large class of epireflective subcategories of Top0 can be 'parametrized' using the interaction of functorial quasi-uniformities with the quasi-uniform bicompletion, we show that using functorial approach structures together with the approach bicompletion developed in [10], only Top0 itself can be retrieved in this way.

Idempotency of Extensions via the Bicompletion

“…Surprisingly, it turns out that a T-section is upper K-true if and only if it satisfies the analogous upper-stability condition (Proposition 2), but that a lower K-true T-section need not be lower K-stable. We shall indicate how these facts mirror certain results for uniform spaces that were obtained in [6] and [10].…”

“…Here we have to mention that the properties of the above class M F have natural analogues in the theory of uniform spaces; see [6, Proposition 5.1 et seq.]. When comparing the quasi-uniform theory with the results in [6], one has to be aware that all sections of the forgetful functor Unif 0 → Tych have the property that corresponds to the quasi-uniform 'upper K-true' (see [3, p. 60]).…”

“…As in the case of [6], our class M F invites consideration of the order-dual class {Y ∈ QU 0 : Y ≥ FTY}. Hager and Rice [10] proved that in Unif every topologypreserving coreflector is stable under completion.…”

The ZKM | Institute for Music and Acoustics

“…The result was the Sound Dome, installed in the Cube theatre of the ZKM, and the interactive control software Zirkonium (Bru¨mmer, Goßmann, Ramakrishnan and Sturm 2006: 3;Bru¨mmer, Ramakrishnan et al 2007/08; Ramakrishnan, this issue). The Sound Dome in the Cube is composed of 47 loudspeakers that surround the listeners in the shape of a dome.…”

Functorial approach structures

“…In [2,3,4,5,7,8,9] so-called functorial (quasi-)uniformities were extensively studied. A functorial quasi-uniformity, is a functor F : Top → QU (where Top, resp.…”