Abstract. We show that the set $\cup_{j\in J}F^n_j $ is arcwise connected, where $ J$ is a subset of $ \overline{\msbm N} $ and $ F^n_j (n\leq j)$ is the set of semi-Fredholm operators of index $n $ and kernel dimension $ j$. The distance of an arbitray operator to $\cup_{j\in J}F^n_j$ is also determined. We show that dist($T,\cup_{j\in J}F^n_j )= $ dist$ (T, F^n_{j_0})$, where $ j_0 =$ inf$\{j: j \in J\} $. Thus $ F^n_{j_0} $ is dense in $\cup_{j\in J}F^n_j$. We also find the boundary of $\cup_{j\in J}F^n_j$ and $ \overline{\cup_{j\in J}F^n_j } $.

AMS Subject Classification (1991): 47A53

Received March 10, 1997. (Registered under 5786/2009.)