  Inspired by a recent work of Wang-Zhao, in this note we prove that for a
fixed $n$-dimensional closed Riemannian manifold $(M^n, g)$, if an
$\mathrm{RCD}(K, n)$ space $(X, \mathsf{d}, \mathfrak{m})$ is Gromov-Hausdorff
close to $M^n$, then there exists a regular homeomorphism $F$ from $X$ to $M^n$
such that $F$ is Lipschitz continuous and that $F^{-1}$ is H\"older continuous,
where the Lipschitz constant of $F$, the H\"older exponent and the H\"older
constant of $F^{-1}$ can be chosen arbitrary close to $1$. This is sharp in the
sense that in general such a map cannot be improved to being bi-Lipschitz.
Moreover if $X$ is smooth, then such a homeomorphism can be chosen as a
diffeomorphism. It is worth mentioning that the Lipschitz-H\"older continuity
of $F$ improves the intrinsic Reifenberg theorem for closed manifolds with
Ricci curvature bounded below established by Cheeger-Colding. The Nash
embedding theorem plays a key role in the proof.
