  Let $(P_1,...,P_n)$ be an $n$--tuple of projections in a unital
$C^*$--algebra $\aa$. We say $\pn$ is complete in $\aa$ if $\aa$ is the linear
direct sum of the closed subspaces $P_1\aa,...,P_n\aa$. In this paper, we give
some necessary and sufficient conditions for the completeness of $\pn$ and
discuss the perturbation problem and topology of the set of all complete
$n$--tuple of projections in $\aa$. Some interesting and significant results
are obtained in this paper.
