Project/Area Number 
12640114

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Shizuoka University 
Principal Investigator 
OHTA Haruto Shizuoka Univ., Fac. of Education, Professor, 教育学部, 教授 (40126769)

CoInvestigator(Kenkyūbuntansha) 
ONO Jin Shizuoka Univ., Fac.of Engineering, Assist. Professor, 工学部, 助教授 (80115443)
KIYOSAWA Takemitsu Shizuoka Univ., Fac. of Education, Professor, 教育学部, 教授 (40015566)
YAMADA Kohzo Shizuoka Univ., Fac. of Education, Assist. Professor, 教育学部, 助教授 (00200717)
YAMAZAKI Kaori Univ. of Tsukuba, Institute of Mathematics, Assistant, 教育学部, 教授 (80301076)
TAMANO Kenichi Yokohama National Univ., Fac. of Engineering, Professor, 工学部, 教授 (90171892)

Project Period (FY) 
2000 – 2001

Project Status 
Completed (Fiscal Year 2001)

Budget Amount *help 
¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2001: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)

Keywords  continuous function / extension / topological space / Banach space / upper semicontinuous / lower semicontinuous / Cembedded / C^*embedded 
Research Abstract 
1. A topological space X is said to have the property (C^* =C) if every C^*embedded closed set of X is Cembedded. In the extension theory of continuous functions, it is an interesting problem to construct a topological space, preferably a completely regular space satisfying the first axiom of countability, which fails to have the property (C^* = C). Concerning this problem, the following results are proved : (1) There exists a completely regular space X in which every point is a Gdelta, and which does not have the property (C^*=C). (2) The Stone space of the Boolean algebra consisting of regular open sets of a completely regular space which is not weakly normal fails to have the property (C^* = C). (3) The Niemytzki plane has the property (C^* = C). 2. Let C(Y) denote the Banach space of all continuous functions on a locally compact space Y which vanish at infinity. We define upper (lower) semicontinuity of maps to C(Y) and establish certain duality between upper (lower) semicontinuity of maps to C(Y) and upper (lower) semicontinuity of setvalued maps to the space of compact sets of C(Y). As an application, the following results are obtained : (1) A Hausdorff space X is paracompact if and only if for every locally compact space Y and every upper semicontinuous map f ; X → C(Y), there exists a continuous map g : X → C(Y) such that f(x) < g(x) for each x in X. (2) Every pointfinite open cover of a topological space X is normal if and only if for every discrete space Y and every two maps g and h from X to C(Y) such that g is upper semicontinuous, h is lower semicontinuous and g 【less than or equal】 h, there exists a continuous map f such that g 【less than or equal】 f 【less than or equal】 h.
