  The self-closeness number $N\mathcal{E}(X)$ of a space $X$ is the least
integer $k$ such that any self-map is a homotopy equivalence whenever it is an
isomorphism in the $n$-th homotopy group for each $n\le k$. We discuss the
self-closeness numbers of certain non-simply-connected $X$ in this paper. As a
result, we give conditions for $X$ such that
$N\mathcal{E}(X)=N\mathcal{E}(\widetilde{X})$, where $\widetilde{X}$ is the
universal covering space of $X$.
