Generalized selection theorems without convexity

In this paper, we extend new selection theorems for almost lower semicontinuous multifunctions TT on a paracompact topological space XX to general nonconvex settings. On the basis of the Kim–Lee theorem and the Horvath selection theorem, we first show that any a.l.s.c.   CC-valued multifunction admits a continuous selection under a mild condition of a one-point extension property. Finally, we apply a fundamental selection theorem, due to Ben-El-Mechaiekh and Oudadess, to modify our selection theorems by adjusting a closed subset ZZ of XX with its covering dimension dimXZ≤0dimXZ≤0. The results derived here generalize and unify various earlier ones from classic continuous selection theory.