Function space topologies between the uniform topology and the Whitney topology

Autor: A. Di Concilio
Rok vydání: 2020
Předmět:
Zdroj: Topology and its Applications. 277:107230
ISSN: 0166-8641
DOI: 10.1016/j.topol.2020.107230
Popis: This paper, dedicated to new function space topologies between the uniform topology and the Whitney topology also in the setting of the ω μ -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 T 2 -compactification of X. And next, when X is a local proximity space, i.e. densely embedded in a natural T 2 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 T 2 -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 T 2 -compactifications of X. Also, whenever X is not locally compact, weak Whitney topologies associated with different T 2 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 ω μ -metric spaces, where ω μ 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 ω μ -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 ω μ -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 ω μ -sequence converging to zero in the order topology. By a proof strategy essentially based on zero-dimensionality of any ω μ -metric space with μ > 0 , we achieve, among others, the following result: Whenever X is an ω μ -additive and paracompact space and ( Y , ρ , G ) is an ω μ -metric space, then the Whitney topology on C ( X , Y ) is independent of the ω μ -metric ρ. More precisely, the Whitney topology is a topological character as in the classical metric case, μ = 0 , and X paracompact.
Databáze: OpenAIRE
Externí odkaz: