We investigate the relationships between the space X and the hyperspaces concerning admissibility and connectedness im kleinen. The following results are obtained: Let X be a Hausdorff continuum, and let A, $B{in}C(X)$ with $A{subset}B$. (1) If X is c.i.k. at A, then X is c.i.k. at B if and only if B is admissible. (2) If A is admissible and C(X) is c.i.k. at A, then for each open set U containing A there is a continuum K and a neighborhood V of A such that $V{subset}IntK{subset}K{subset}U$. (3) If for each open subset U of X containing A, there is a continuum B in C(X) such that $A{subset}B{subset}U$ and X is c.i.k. at B, then X is c.i.k. at A. (4) If X is not c.i.k. at a point x of X, then there is an open set U containing x and there is a sequence ${S_i}^{infty}_{i=1}$ of components of $ar{U}$ such that $S_i{longrightarrow}S$ where S is a nondegenerate continuum containing the point x and $S_i{cap}S={emptyset}$ for each i = 1, 2, ${cdots}$.