  Let $f$ be an $R$-closed homeomorphism on a connected orientable closed
surface $M$. In this paper, we show that If $M$ has genus more than one, then
each minimal set is either a periodic orbit or an extension of a Cantor set. If
$M = \mathbb{T}^2$ and $f$ is neither minimal nor periodic, then either each
minimal set is finite disjoint union of essential circloids or there is a
minimal set which is an extension of a Cantor set. If $M = \mathbb{S}^2$ and
$f$ is not periodic but orientation-preserving (resp. reversing), then the
minimal sets of $f$ (resp. $f^2$) are exactly two fixed points and other
circloids and $\mathbb{S}^2/\widetilde{f} \cong [0, 1]$.

Dynamical Systems