幾乎下半連續多值函數的連續選擇

Continuous Selections For Almost Lower Semicontinuous Multifunctions

黃建豪(Chien-Hao Huang)

指導教授 ： 朱亮儒

國立臺灣師範大學/理學院/數學系/碩士(2009年)

摘要

這篇論文裡，在非緊緻、非凸集的情況下，我們將會得到一些關於定義在paracompact拓樸空間上，幾乎下半連續的多值函數T之連續選擇定理。我們考慮三種較感興趣的主題；每一種主題都處理了廣泛類型的連續選擇問題。一種是介紹且分析一些已知的連續選擇問題。基於Deutsch-Kenderov定理及equicontinuous性質，首先即使在T沒有下半連續，我們證明T只要是幾乎下半連續的一般化的連續選擇定理。第二，我們證明一些抽象化凸性與連續選擇性質之間的關係。在加入one point extension性質的調整下，我們證明在一個metric space裡即使沒有凸性，在賦予C-set結構的情況下，仍然會有連續選擇的性質。最後，在改變X中一個covering dimension小於或等於0的閉集Z的情況下，我們將調整我們的連續選擇定理。而在此導出的結果一般化且一致化了很多早期典型的連續選擇定理。

關鍵字

連續選擇 ； 近似連續選擇 ； 下半連續 ； 單位分割 ； 幾乎下半連續 ； 等度連續性質 ； C-空間 ； C-集合 ； LC-賦距空間 ； 單點延拓性質 ； 覆蓋維度

並列摘要

In this paper, we obtain several new continuous selection theorems for almost lower semicontinuous multifunctions T on a paracompact topological space X, in the general noncompact and/or nonconvex settings. We consider three interesting topics in the selection theory; each of these topics deals with a broad class of selection problems. One is to introduce and analyze some well known selection theorems. Based on Deutsch-Kenderov theorem and an equicontinuous property, we first establish a generalized
selection theorem for the multifunctions, even without requiring lower semicontinuity on T, but merely an almost lower semicontinuous multifunction. Secondly, we establish some relationships between abstract convexity and the selection property. Under a mild condition of one point extension property, we show that a C-set structure on a metric space without convexity still has the continuous selection property. Finally, we modify our selection
theorems by adjusting a closed subset Z of X with its covering dimension less or equal to 0. These results derived here generalize and unify various earlier ones from classic continuous selection theory.

並列關鍵字

continuous selection ； ε-approximate selection ； lower semicontinuous ； partition of unity ； almost lower semicontinuous ； equicontinuous property ； C-space ； C-set ； LC-metric space ； one point extension property ； covering dimension

參考文獻

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[4] Z. Chen (1988). An equivalent condition of continuous metric selection, J. Math. Anal. Appl. 136, 298-303.

[5] F. Deutsch and P. Kenderov (1983). Continuous selections and approximate selection for set-valued mappings and applications to metric projections, SIAM J. Math. Anal. 14, 185-194.