Floquet time crystals, which break the discrete-time translational symmetry, are an interesting phenomenon in non-equilibrium systems. Understanding the stiffness and robustness of discrete-time crystal (DTC) phases in many-body systems is important, and finding an accurately solvable model can pave the way to understanding DTC phases. Here we propose and study a solvable spin chain model by mapping it to Floquet superconductors via the Jordan-Wigner transform. The state diagram of a Floquet topological system is characterized by topological invariants and shows the presence of anomalous edge states. Subharmonic oscillation, a typical DTC signal, is generated from such edge conditions and protected by the topology. We also examine the robustness of the DTC by adding symmetry-maintaining and symmetry-breaking perturbations. Our results on topologically protected DTCs can provide a deeper understanding of DTCs when generalized to other interacting or dissipative systems.