  We will show that a system is synchronized if and only if it has a cover
whose cover map is semi-open. Also, any factor code on an irreducible sofic
shift is semi-open and the image of a synchronized system by a semi-open code
is synchronized. On the other side, right-closing semi-open extension of an
irreducible shift of finite type is of finite type. Moreover, we give
conditions on finite-to-one factor codes to be open and show that any semi-open
code on a synchronized system is bi-continuing a.e.. We give some sufficient
conditions for a right-continuing a.e. factor code being right-continuing
everywhere.
