A set is called r-closed left-r.e. iff every set r-reducible to it is also a left-r.e. set. It is shown that some but not all left-r.e. cohesive sets are many–one closed left-r.e. sets. Ascending reductions are many–one reductions via an ascending function; left-r.e. cohesive sets are also ascending closed left-r.e. sets. Furthermore, it is shown that there is a weakly 1-generic many–one closed left-r.e. set. We also consider initial segment complexity of closed left-r.e. sets. We show that initial segment complexity of ascending closed left-r.e. sets is of sublinear order. Furthermore, this is near optimal as for any non-decreasing unbounded recursive function g, there are ascending closed left-r.e. sets A whose plain complexity satisfies C ( A ( 0 ) A ( 1 ) ⋯ A ( n ) ) ⩾ n / g ( n ) for all but finitely many n. The initial segment complexity of a conjunctively (or disjunctively) closed left-r.e. set satisfies, for all ε > 0, for all but finitely many n, C ( A ( 0 ) A ( 1 ) ⋯ A ( n ) ) ⩽ ( 2 + ε ) log ( n ).