  We introduce the notion of a Bishop topological group i.e., a group X
equipped with a Bishop topology of functions F such that the group operations
of X are Bishop morphisms with respect to F. A closed subset in the
neighborhood structure of X induced by its Bishop topology F is defined in a
positive way i.e., not as the complement of an open subset in X. The
corresponding closure operator, although it is not topological, in the
classical sense, does not involve sequences. As countable choice (CC) is
avoided, and in agreement with Richman's critique on the use of CC in
constructive mathematics, the fundamental facts on closed subsets in Bishop
topological groups shown here have a clear algorithmic content. We work within
Bishop's informal system of constructive mathematics BISH, without countable
choice, equipped with inductive definitions with rules of countably many
premises.
