In this paper, through analyzing data from the Solar Dynamics Observatory (SDO) and the Global Oscillation Network Group (GONG), we present a study on the formation of a double-decker filament in NOAA Active Region 12665 from 2017 July 8 to 14. We find that magnetic reconnection occurs between two smaller filaments to form a longer filament. According to the evolution of the leading sunspot, it is obvious that the sunspot experiences a continuous rotation around its umbra. During the period from 03:00 UT on July 11 to 10:00 UT on July 14, the average speed of sunspot rotation is about 3.degrees 7 hr(-1). The continuous rotation of sunspot stretches the filament and results in the formation of a reversed S-shaped filament. Due to the motion of the magnetic field and internal magnetic reconnection, the filament splits into two branches and forms a double-decker filament structure. In the process of filament separation, internal magnetic reconnection can also accelerate the filament separation. Nonlinear force-free field extrapolation indicates that there are two magnetic flux ropes, which are consistent with the observed results. Eventually, the upper filament erupts and produces an M-class flare and a halo coronal mass ejection.