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。

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.

[1] H. Ben-El-Mechaiekh and M. Oudadess (1995). Some selection theorems without convexity, J. Math. Anal. Appl. 195, 614-618.
[2] F. S. de Blasi and J. Myjak (1985). Continuous selections for weakly Hausdroff lower semicontinuous multifunctions, Proc. Amer. Math. Soc. 93, 369-372.
[3] F. E. Browder (1984). Coincidence theorems, minimax theorems, and variational inequalities, Contemp. Math. 26, 67-80.
[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.
[6] F. Deutsch, V. Indumathi and K. Schnatz (1988). Lower semicontinuity, almost lower semicontinuity, and continuous selections for set-valued mappings, J. Approx. Theory. 53, 266-294.
[7] V. G. Gutev (1993). Selections under an assumption weaker than lower semicontinuity, Topol. Appl. 50, 129-138.
[8] C. Horvath (1991). Contractibility and generalized convexity, J. Math. Anal. Appl. 156, 341-357.
[9] H. Kim and S. Lee (2006). Approximate selections of almost lower semicontinuous multimpas in C-spaces, Nonlinear. Anal. TMA 64, 401-408.
[10] E. Michael (1956). Continuous selections I, Ann. Math. 63, 361-382.
[11] E. Michael and C. Pixley (1980). A unied theorem on continuous selections, Pacific. J. Math. 87, 187-188.
[12] K. Przeslawinski and L. E. Rybinski (1990). Michael selection theorem under weak lower semicontinuity assumption, Proc. Amer. Math. Soc. 109, 537-543.
[13] D. Repoves and P. V. Semenov (1998). Continuous Selections of Multivalued Mappings, Kluwe Academic Publishers.
[14] R. T. Rockafellar (1970). Convex Analysis, Princeton, New Jersey, Princeton University Press.
[15] X. Wu and S. Shen (1996). A further generalization of Yannelis-Prabhakar's continuous selection theorem and its applications, J. Math. Anal. Appl. 197, 61-74.
[16] Y. Zhang (2003). On the problem of continuous selection for almost lower semicontinuous set-valued mapping, Journal of Jilin University, science edition 41, 304-308.
[17] X. Zheng (1997). Approximate selection theorems and their applications, J. Math. Anal. Appl. 212, 88-97.