関数空間の一様構造及び連続条件

Using uniform structures, Dieudonne [3] systematized a number of results on topologies for homeomorphism groups which had been published till 1947. Since then, as to properties on continuity, only sufficient conditions have been given under special uniformities.
Let X be a set, Y be a uniform space endowed with a uniform structure U, S be a family of subsets of X, and F be the family of all mappings of X into Y. For each set A∈S and each entourage U∈U, let W(A, U) denote the set of all pairs (u, v) of mappings of X into Y such that (u (x), v (x)) ∈U for all x∈A. Then {W(A, U)│A∈S, U∈U} form a fundamental system of entourages of a uniformity W on F under the proper conditions on S (Theorem 1).
The purpose of this paper is to find the most general conditions possible, expressed by the properties of S and U, that satisfy the following basal conditions on continuity with respect to the uniformity W : i) the mapping (u,x) →u(x) of C × X into Y is continuous, where X is a topological space and C is a family of continuous mappings of X into Y (Theorems 2 and 3), ii) the mapping (u, v) → uv of C X C into C is continuous, where X and Y are the same uniform space and CC⊂C (Propositions 4 and 5 ; Theorems 4 and 5). These are the basal conditions often required to be satisfied for semigroups of continuous transformations of a uniform space.
From our results, it is conjectured that if a uniformity W on F satisfies these basal conditions for the family of all continuous mappings of a space into itself which has several properties similar to those of euclidean spaces, then W must be the uniformity of compact convergence. In fact it is affirmative (cf. Karube [5]).
For topological terms and notations we follow the usage of N. Bourbaki [2].