Topology of the Semigroup of Singular Endomorphisms

_n

of singular endomorphisms of an n-dimensional vector space over K are discussed here. Since S _n is known to be an idempotent generated regular semigroup, we pay more attention to the topological properties of the set E _n of idempotents in S _n . The local structure of E _n is shown to be that of a C ^infinity-manifold and of a finite-dimensional vector bundle over the Grassmann manifolds. The topology of the biorder relations and sandwich sets are also discussed.