Electrons in zero external magnetic field can be studied with the Kohn-Sham (KS) scheme of either density functional theory (DFT) or spin-DFT (SDFT). The latter is normally used for open-shell systems because its approximations appear to model better the exchange and correlation (xc) functional, but also because, so far the application of DFT implied a closed-shell-like approximation. In the first part of this Communication, we show that correcting this error for open shells allows the approximate DFT xc functionals to become as accurate as those in SDFT. In the second part, we consider the behavior of SDFT for zero magnetic field. We show that the KS equations of SDFT emerge as the generalized KS equations of DFT in this limit, thus establishing a so far unknown link between the two theories.