Local Compactifications Research Articles

Function space topologies between the uniform topology and the Whitney topology

This paper, dedicated to new function space topologies between the uniform topology and the Whitney topology also in the setting of the ωμ-metric spaces, splits in two parts. In the former, where X is a Tychonoff space and (Y,d) is a non-discrete metric space, we explore suggestive uniformizable function space topologies on C(X,Y), the set of all continuous functions from X to Y, located between the uniform topology and the Whitney topology. In the Whitney uniformity, whose natural associated topology is the Whitney topology, any continuous function from X to the positive reals gives a measure of closeness between functions in C(X,Y). But, a less stringent and, by the way, efficient uniform control can be performed equally well by limiting, as for example at a first glance, to the measures deriving from all continuous positive functions continuously extendable to a T2-compactification of X. And next, when X is a local proximity space, i.e. densely embedded in a natural T2 local compactification l(X), by limiting to the positive ones in C(l(X),R). We investigate two classes of Tychonoff spaces. That of locally compact ones splittable in two essentially different cases: X hemicompact or not. And, that of spaces densely embedded in a locally compact one. We prove that, whenever X is hemicompact, then any weak Whitney topology relative to a T2-compactification of X agrees with the classical one. Whenever X is locally compact but not hemicompact, then the weak Whitney topology associated with its one-point compactification reduces just to the uniform topology. In the case X is locally compact, paracompact but not hemicompact, thus the free union of an uncountable family of open σ-compact subsets, then, between the uniform topology and the Whitney topology there is a great variety of weak Whitney topologies relative to T2-compactifications of X. Also, whenever X is not locally compact, weak Whitney topologies associated with different T2 local compactifications of X are generally different as is the case if X is the rational Euclidean line. So, weakening the Whitney topology but without renouncing to the uniform convergence, we produce different uniformizable topologies on C(X,Y) related to various significant structures on X.In the latter, since ωμ-metric spaces, where ωμ is an ordinal number, fill a large and attractive class of peculiar uniform spaces containing the usual metric ones, we focus our attention on the ωμ-metric framework. Indeed, we extend the Whitney topology to C(X,Y), where X is again a Tychonoff space but Y is replaced with an ωμ-metric space. Precisely, the range space Y carries a distance ρ:Y×Y→G, sharing the usual formal properties with real metrics but valued in an ordered Abelian additive group G, which admits a strictly decreasing ωμ-sequence converging to zero in the order topology. By a proof strategy essentially based on zero-dimensionality of any ωμ-metric space with μ>0, we achieve, among others, the following result: Whenever X is an ωμ-additive and paracompact space and (Y,ρ,G) is an ωμ-metric space, then the Whitney topology on C(X,Y) is independent of the ωμ-metric ρ. More precisely, the Whitney topology is a topological character as in the classical metric case, μ=0, and X paracompact.

Boolean Algebra of One-Point Local Compactifications

Boolean Algebra of One-Point Local Compactifications

Open and other kinds of map extensions over zero-dimensional local compactifications

Generalizing a theorem of Ph. Dwinger (1961) [7], we describe the partially ordered set of all (up to equivalence) zero-dimensional locally compact Hausdorff extensions of a zero-dimensional Hausdorff space. Using this description, we find the necessary and sufficient conditions which has to satisfy a map between two zero-dimensional Hausdorff spaces in order to have some kind of extension over arbitrary, but fixed, Hausdorff zero-dimensional local compactifications of these spaces; we consider the following kinds of extensions: continuous, open, quasi-open, skeletal, perfect, injective, surjective, dense embedding. In this way we generalize some classical results of B. Banaschewski (1955) [1] about the maximal zero-dimensional Hausdorff compactification. Extending a recent theorem of G. Bezhanishvili (2009) [2], we describe the local proximities corresponding to the zero-dimensional Hausdorff local compactifications.

Extensions of Lévy–Khintchine formula and Beurling–Deny formula in semi-Dirichlet forms setting

The Lévy–Khintchine formula or, more generally, Courrège's theorem characterizes the infinitesimal generator of a Lévy process or a Feller process on R d . For more general Markov processes, the formula that comes closest to such a characterization is the Beurling–Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Lévy–Khintchine and Beurling–Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric.

Theory of convex local compactifications with applications to lebesgue spaces

A general theory is developed to imbed continuously and densely a topological space with a boundedness into a �-compact (possibly also locally compact and closed) convex subset of a locally convex space. A class of such convex �-compactifications of the Lebesgue space L p (;IR n ) is then studied, and particular cases are identified with some known ex- tensions, e.g. with Young-Schonbek or DiPerna-Majda measures. It enables a thorough investigation of their topological/geometrical properties and relations between them. Affine continuous extensions of Nemytskii mapping are systematically studied, as well.

Quasi-homeomorphisms of Dirichlet forms

Extending fundamental work of M. Fukushima, M. L. Silverstein, S. Carrillo Menendez, and Y. Le Jan (cf. [F71a, 80], [Si74], [Ca-Me75], [Le77]) it was recently discovered that there is a one-to-one correspondence between (equivalent classes of) all pairs of sectorial right processes and quasi-regular Dirichlet forms (see [AM91], [AM92], [AMR90], [AMR92a], [AMR92b], [MR92]). Based on the potential theory for quasi-regular Dirichlet forms, it was shown that any quasi-regular Dirichlet form on a general state space can be considered as a regular Dirichlet form on a locally compact separable metric space by “local compactification”. There are several ways to implement this local compactification. One relies on h-transformation which was mentioned in [AMR90, Remark 1.4]. A direct way using a modified Ray-Knight compactification was announced on the “5th French-German meeting: Bielefeld Encounters in Mathematics and Physics IX. Dynamics in Complex and Irregular Systems”, Bielefeld, December 16 to 21, 1991, and the “Third European Symposium on Analysis and Probability”, Paris, January 3-10, 1992, and appeared in [MR92, Chap. VI] and [AMR92b] (see also the proof of Theorem 3.7 below). One can also do this by Gelfand-transform. This way was found by the first named author independetly and announced in the “12th Seminar on Stochastic Processes”, Seattle, March 26-28, 1992. It will be discussed in Section 4.

Dyonic bound states from local compactification.

Reflecting the breakdown of continuous duality in Kaluza-Klein theories, higher-dimensional geodesics cannot generically represent dyonic trajectories. The conjectured internal structure of dyons can be constructively realized via a bound state of spin eg, owing its existence to the fact that the extra dimensional scale, while shrinking near an elementary electric source, actually explodes when approaching a light magnetic monopole.

Convergence of minima in convergence spaces1

Sufficient and necessary conditions for the persistence and the adherence of minima in general convergence spaces are provided and specialized in the cases of topological and sequential convergences, as well as in the case of local compactifications. In terms of multifunctions, the inquired stability properties of minima amount to lower semicontinuity (persistence) and to graph-closedness (adherence).

Local compactification and black holes in d = 11 supergravity

We consider solutions of d = 11 supergravity describing a product of a four-dimensional Schwarzschild geometry and a seven-sphere whose radius R does depend on r. Three cases are studied as follows. 1. (i) Vanishing photon fields: an exact solution with r-dependent R( r) is presented which exhibits a horizon. 2. (ii) Freund-Rubin asymptotics: Numerical results show that in this case solutions exist without a horizon at a finite value of r, but with R(0) = ∞. 3. (iii) Englert asymptotics: a striking analogy with the 't Hooft-Polyakov monopole is found which suggests the existence of a completely regular solution which interpolates between the Englert solution at r = ∞ and the Freund-Rubin solution at r = 0.