\(\text{End}(V)\) is the semigroup of all endomorphisms of an \(n\)-dimensional vector space \(V\) over either the field \(\mathbb{R}\) of real numbers or the field \(\mathbb{C}\) of complex numbers. The subsemigroup of \(\text{End}(V)\) consisting of all singular endomorphisms is denoted by \({\mathbf S}_n\) and it is well known that \({\mathbf S}_n\) is generated by \({\mathbf E}_n\), its set of idempotents. \(\text{End}(V)\) is identified with \(M_n(\mathbb{K})\), the semigroup of all \(n\times n\) matrices over \(\mathbb{K}\) where either \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{K}=\mathbb{C}\) and consequently, \(\text{End}(V)\) can be regarded as a Euclidean topological space. As such, \({\mathbf S}_n\), though not a manifold, is a closed subspace of \(\text{End}(V)\) and is therefore a complete topological semigroup. Since \({\mathbf E}_n\) generates \({\mathbf S}_n\) it is an important subspace and, consequently, receives a good deal of attention. For example, denote by \(E(k)\), \(0\leq k\leq n-1\), the set of idempotents of rank \(k\). It is shown that these sets are precisely the path components of \({\mathbf E}_n\). Furthermore, it is shown that each \(E(k)\) is a \(C^\infty\)-manifold of dimension \(2k(n-k)\). Results are obtained about biorder relations and sandwich sets on \({\mathbf E}_n\), concepts introduced by the second author [in: Structure of regular semigroups I (Mem. Am. Math. Soc. 224) (1979; Zbl 0457.20051)]. The authors conclude by examining \(E(1)\), the space of all nonzero singular idempotent endomorphisms of the two-dimensional real vector space. They note that \(E(1)\) can be embedded in a three-dimensional real vector space and as such is a hyperboloid of one sheet whose principal section is the set of all selfadjoint idempotents in \(E(1)\).