Topologicality of the Whitney topology on function spaces in the ωμ-metric framework

Abstract

Let X be a Tychonoff space, (Yd) a metric space and C(X,Y) the set of all continuous functions from X to Y. In the classical Whitney topology on C(X,Y) the approximation of functions is done by the means of continuous positive functions from X to the Euclidean real line. It is known that, whenever X is paracompact, the Whitney topology is independent of the metric d, in fact being a topological character. An analogous result holds whenever (Y,d) is replaced by an ωμ-metric space (Y,ρ,G), where ωμ is a regular ordinal number, ρ:Y×Y→G is a distance sharing the usual formal properties with real metrics but valued in an additive totally ordered Abelian group G admitting a strictly decreasing ωμ-sequence converging to zero in the order topology, or, the same, of character ωμ. Precisely, whenever ωμ is an uncountable ordinal and X is paracompact and ωμ-additive, then the Whitney topology on C(X,Y) is independent of the ωμ-metric ρ, in fact being again a topological character. The purpose of this paper is to fill the gap in the case the basic groups are of countable character but eventually non Archimedean. It comes up as very natural the preliminary question: how does a totally ordered Abelian group G look like? The Hahn's Embedding Theorem and the Hahn's Completeness Theorem provide models so making easier the investigation of their order topology in the distinguished two cases of countable character: the basic group G has finitely many Archimedean classes or infinitely many, then countably many, Archimedean classes. In the former, any basic group G with n,n≥2, Archimedean classes can be identified up to a bijective monotone homomorphism with a topological subgroup of Rn, which is a one-dimensional topological group when equipped with the pointwise addition and the lexicographic absolute value metric. In the latter, any basic group G with countably many Archimedean classes can be identified up to a bijective monotone homomorphism with a topological subgroup of Rℵ0, the set of all real sequences, which is a zero-dimensional topological group when equipped with the pointwise addition and the lexicographic absolute value metric. In the former case, we establish the following result including the classic metric one: Let X be paracompact. If(Y,ρ)is a G-metric with G of countable character and with finitely many Archimedean classes, eventually a real metric space, then the Whitney topology onC(X,Y)relative to ρ and G is independent of both ρ and G, in fact being a topological character. And in the latter one, we achieve the general following result: Let X be paracompact and(Y,ρ)a G of countable character and with infinitely many Archimedean classes. Then, the Whitney topology onC(X,Y), when different from the relative uniform topology, is independent of both ρ and G, so agreeing with the classical one, if and only if any real valued positive continuous function on X can be minorized with a positive locally constant function.1