  The purpose of this paper is to present, for all $n\ge 3$, very simple
examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds
$M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set
$S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be
approximated by an embedding, i.e. $f$ is a {\sl near-embedding}.
