  In this paper, we prove first that the space of minimal sets of any
homeomorphisms $f:X\to X$ of a regular curve $X$ is closed in the hyperspace
$2^X$ of closed subsets of $X$ endowed with the Hausdorff metric, and the
non-wandering set $\Omega (f)$ is equal to the set of recurrent points of $f$.
Second, we study the continuity of the map $\omega_f:X\to 2^X;x\mapsto \omega_f
(x) $, we show for instance the equivalence between the continuity of $\omega_f
$ and the equality between the $\omega$-limit set and the $\alpha $-limit set
of every point in $X $. Finally, we prove that there is only one (infinite)
minimal set when there is no periodic point.
