  The notion of an open quantum subgroup of a locally compact quantum group is
introduced and given several equivalent characterizations in terms of
group-like projections, inclusions of quantum group C*-algebras and properties
of respective quantum homogenous spaces. Open quantum subgroups are shown to be
closed in the sense of Vaes and normal open quantum subgroups are proved to be
in 1-1 correspondence with normal compact quantum subgroups of the dual quantum
group.
