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タイトルヨミ
カンスウ クウカン ノ イチヨウ コウゾウ オヨビ レンゾク ジョウケン
日本語以外のタイトル
Uniformities for Function Spaces and Continuity Conditions
ファイル
言語
英語
著者
苅部 孝
内容記述(抄録等)
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].
掲載誌名
島根大学教育学部紀要. 自然科学
2
開始ページ
1
終了ページ
12
ISSN
05869943
発行日
1968-12-28
NCID
AN00107941
出版者
島根大学教育学部
出版者別表記
The Faculty of Education Shimane University
資料タイプ
紀要論文
部局
教育学部
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