The question itself seems too elementary for this site, since it just involves the standard axiomatic treatment of root systems as in Bourbaki Groupes et algebres de Lie, VI.1.7. The question is really about an arbitrary reductive algebraic group (with nontrivial derived group) over an algebraically closed field, along with its Borel subgroups in natural bijection with systems of positive roots relative to a fixed maximal torus. Such a torus $T$ lies in exactly $|W|$ Borel subgroups, where $W=N_G(T)/T$ is the Weyl group. At this point the axiomatic theory takes over and provides straightforward criteria for a given "closed" set of roots to be the positive roots for some choice of simple roots: the set has to be disjoint from its negative and together with its negative exhaust all roots (Prop. 20, Cor. 1).
The question itself seems too elementary for this site, since it just involves the standard axiomatic treatment of root systems as in Bourbaki Groupes et algebres de Lie, VI.1.7. The question is really about an arbitrary reductive algebraic group (with nontrivial derived group) over an algebraically closed field, along with its Borel subgroups in natural bijection with systems of positive roots relative to a fixed maximal torus. Such a torus $T$ lies in exactly $|W|$ Borel subgroups, where $W=N_G(T)/T$ is the Weyl group. At this point the axiomatic theory takes over and provides straightforward criteria for a given "closed" set of roots to be the positive roots for some choice of simple roots.