  The present work considers the properties of classes of generally convex sets
in the plane known as $1$-semiconvex and weakly $1$-semiconvex. More
specifically, the examples of open and closed weakly $1$-semiconvex but non
$1$-semiconvex sets with smooth boundary in the plane are constructed. It is
proved that such sets consist of minimum four connected components. In
addition, the example of closed, weakly $1$-semiconvex, and non $1$-semiconvex
set in the plane consisting of three connected components is constructed. It is
proved that such a number of components is minimal for any closed, weakly
$1$-semiconvex, and non $1$-semiconvex set in the plane.
