We performed a systematic study of the Doppler shifts and electron densities measured in an EUV bright point (hereafter BP) observed in more than 10 EUV lines with formation temperatures from log (T/K) = 4.5 to 6.3. Those parts of a BP seen in transition region and coronal lines are defined as its cool and hot components, respectively. We find that the transition from cool to hot occurs at a temperature around log (T/K) = 5.7. The two components of the BP reveal a totally different orientation and Doppler-shift pattern, which might result from a twist of the associated magnetic loop system. The analysis of magnetic field evolution and topology seems to favor a two-stage heating process, in which magnetic cancellation and separator reconnection are powering, respectively, the cool and hot components of the BP. We also found that the electron densities of both components of the BP are higher than those of the surrounding quiet Sun, and comparable to or smaller than active region densities.