連続関数の空間の連続性について
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(2) Journal of Hokkaido University of Education (Section II A) Vol. 31, No. 2 Bgiffi 56 ^ 3 H ^WSfiii±sf:^ (H 2 g&A) ^ 31^ ^ 2-^- March, 1981. A Note on Completeness of the Space of Continuous Functions. Akio TSUBOUCHI Mathematics Laboratory, Asahikawa College, Hokkaido University of Education, Asahikawa 070. m Hg^: M^m^sr^»?^^ »M^W;WiSJi|^Wf^. Abstract Let E be an arbitrary topological space and C(E) be the space of all real-valued and bounded continuous functions on E. H. Nakano has argued that a sufficient condition for C(E) to be ocomplete (every open Fo--set has an open closure) is that E is cr-universal.. In this paper, we shall show that this condition is not necessary. And we shall remark the necessary and sufficient condition for C(E) on an arbitrary topological space E to be o'-complete (although it is not so in pure topological form).. § 1. Negative Example We let E={ai 02, •••, an, •••}U{bi, 62}, and introduce a topology to E by the following. family of closed sets :. (1) E and ^ (2) A={ai,a2,---,an,---}U{bi}. (3) Finite subsets of A (4) Finite subsets of E containing {61,62} Though [ai, 02, •", an, •••} is a o'-open set, its closure is A. Namely, E is not o-universal.. As every closed set containing bz contains bi, we have /(6i)=/(&2) for every function / £ C(E). There exist no open sets in {ai, a^-", an,"-} that are not empty and dis joint. Furthermore {fli, az,"-, an,-"} is not an open-closed set. Therefore / is constant for every / £ C(E).. (61).
(3) Akio TSUBOUCHI. 2. Necessary and Sufficient Condition Theorem. Let E be an arbitrary topological space and C(E) be the space of all real-valued and bounded functions on E. C(E) is a continuous semi-ordered linear space if and only if E satisfies the following conditions : 1) there exists the smallest open-closed set Uf containing. Sf={x\\/f^C{E){f(x)>0)}. 2) if SfC}Sg=(/> for any /, g £ C(£), thfen ^/ H ^=<^. 3) all open-closed sets of E constitute a cr-complete lattice.. Proof. Suppose C(E) is o-complete, then by the completeness of C(E) any non-negative continuous function / is decomposed by the constant function 1 as follows. 1=[/]1+(1-[/])1. Here [/]1=U(1 H n\f\}. As [/]ln (1-[/]1)=0, [/]1 is a characteristic function Xy for n=\. some open-closed set V, and |/| n (!—[/] )1=0 implies Vc Sf. The fact that f± x uc for all open-closed set U such that Sf c U shows [f]l=Xv-LXuc. This implies V C U, namely U is the smallest open-closed set containing Sf. If Sf^Sg=(/) for every non-negative function /,^-£C(E), then we have [/]lJ-[,g"]l by f±g. Here Xu^=[f]l±[g]l=Xug , so we have Uf n L^=<^. Let Un (%=!, 2, •••) be any sequence of open closed sets in E, then %un (= C(E) (%=!, 2, •••) has an infimum /o=H Xun by assumption. Therfore Ufo is an infimum of L7n (%=1,2, •••) in n=l. all of open-closed sets of E. Q. E. D.. References. [1] Nakano, H. (1941), Uber das System aller stetigen Funktionen auf einem topologischen Raum. Proc. Imp. Acad. Tokyo, Vol. 17, 308-310.. [2] Nakano, H. (1950), Modern spectral theory, Tokyo Mathematical Book Series, Vol. 2, Tokyo.. (62).
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