After reading the chapter "Strings with world-sheet supersymmetry" in Becker, Becker, Schwarz book multiple times, I am still confused about the following things.
1) Open string doubt
They prove that, in order for the boundary term in the variation of the action to vanish, one has to impose EITHER Ramond condition OR Neveu-Schwarz condition, which lead to different mode expansions. After quantization and GSO projection, they then analyse the spectrum and find that the number of bosons in NS sector is the same as the number of fermions in the R sector, in both the ground state and first excited level. They then conclude that the theory is supersymmetric.
If one has to choose only one of the two mentioned condition, how can one have both spectra? How can the theory really be supersymmetric? I would say that it is just a curious feature of the theory, but I clearly miss their point.
2) Closed string doubt
As far as I understood, type II theories are closed string theories. At page 137, they again state what happens in each "sector" in both type IIA and type IIB. If I, for example, choose the NS-NS sector (which is the same for both theories), it would seem that there are no fermions in the theory.
What am I missing again?