Improvements of Some Principal Results in Abstract Duality

Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this note, we would like to establish some improved results, which are basic principle in analysis, of the classical Ascoil theorems and give extremely important results of convenient consequences of these results.


Introduction
Let Ω be a topological space and X = (X, U) a uniform space with the uniformity U ⊂ 2 (X 2 ) .For S ⊂ C(Ω, X) , the topology τ u of uniform convergence on compact sets is equal to the compact open topology τ k ([2, p.284]).Then we have three classical Ascoil theorems as follows.
Theorem A. Let Ω be locally compact, X a uniform space and S a closed set in (C(Ω, X), τ k ).Then S is compact in (C(Ω, X), τ k ) if and only if S is equicontinuous and {g(ω) : [3,Ch.7]).

Theorem B.
Let Ω be a Hausdorff k space or regular k space, X a Hausdorff uniform space and S ⊂ C(Ω, X). 3,Ch.7 ]).
Theorem C. Let Ω be a compact space, X a pseudometric space and S ⊂ C(Ω, X).Let uΩ be the topology on C(Ω, X) for which f α uΩ / / f if and only if Here comes two problems.Firstly, a compact space might not be locally compact, and there exist many non-regular compact spaces and non-Hausdorff compact spaces.In fact, if a compact space is not locally compact, then it is neither regular nor Hausdorff.Besides, many uniform spaces are not metrizable while Theorem C is dealing with the pair (Ω, X) of compact Ω and metrizable X. Hence we would like to establish a result on a level with Theorem A and B for the pair (Ω, X) of compact space Ω and uniform space X .
Secondly, for the simplest case of X = C , Theorem C has an improved version named Arzela-Ascoli theorem: for compact space Ω and S ⊂ C(Ω, C), S is relatively compact in (C(Ω, C), uΩ) if and only if S is equicontinuous and {g(ω) : g ∈ S, ω ∈ Ω} is relatively compact in C, i.e., sup g∈S,ω∈Ω |g(ω)| < +∞ ([4, p.266]).Hence we would like to establish a result on a level with the Arzela-Ascoli theorem for the pair (Ω, X) of compact space Ω and pseudometric space X.
Observe that if S ⊆ F ⊆ X Ω , then We are now in a position to state and prove the main theorem.
Conversely, suppose that X is Hausdorff and S (F,σΩ) is compact in (F, σΩ).