  For a closed connected oriented manifold $M$ of dimension $2n$, it was proved
by M\o ller and Raussen that the components of the mapping space from $M$ to
$S^{2n}$ have exactly two different rational homotopy types. However, since
this result was proved by the algebraic models for the components, it is
unclear whether other homotopy invariants distinguish their rational homotopy
types or not. The self-closeness number of a connected CW complex is the least
integer $k$ such that any of its self-map inducing an isomorphism in $\pi_*$
for $*\le k$ is a homotopy equivalence, and there is no result on the
components of mapping spaces so far. For a rational Poincar\'e complex $X$ of
dimension $2n$ with finite $\pi_1$, we completely determine the self-closeness
numbers of the rationalized components of the mapping space from $X$ to
$S^{2n}$ by using their Brown-Szczarba models. As a corollary, we show that the
self-closeness number does distinguish the rational homotopy types of the
components. Since a closed connected oriented manifold is a rational Poincar\'e
complex, our result partially generalizes that of M\o ller and Raussen.
