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.