Topological superconductivity (TS) has been discussed widely in many superconducting systems with different symmetries, such as time-reversal symmetry, mirror symmetry, inversion symmetry, etc. In this paper we focus on the effects of the combined space-time inversion symmetry $PT$ on the gapped TS. By using Clifford algebra, we find that only one-dimensional (1D) and two-dimensional (2D) superconducting systems with $C^2={\left(PT\right)}^2=1$ ($C$ is particle-hole operator) can be topologically nontrivial, different from the conventional superconducting classifications. Starting from the minimum Dirac model, we construct the representative 1D and 2D models, which only show topology with $PT$ symmetry. These nontrivial models are called Stiefel-Whitney TS, for which the Hamiltonian and wave functions of the system can be real up to a basis choice. In terms of the 2D model, Stiefel-Whitney TS not only corresponds to the second-order TS having zero energy corner modes, but also can show topologically protected nodal points in the Brillouin zone. The possible experimental realizations for 1D and 2D Stiefel-Whitney TS from superconducting proximity effects are pointed out. We expect that these theoretical results will arouse the explorations on Stiefel-Whitney TS.

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• Proximity effect
• Symmetry protected topological states
• Topological phase transition
• Topological superconductors
• Bogoliubov-de Gennes equations
• PT-symmetry