How can a person sleep with their eyes open?

REM sleep: The human sleep cycle has different phases. There are two major ones called rem sleep and non-rem sleep. Rem stands for rapid eye movements. During this phase, the person's eyes are open. This phase is followed by the non-rem sleep when the person's body systems "quiet" down and he appears to have a more relaxed and peaceful "normal" sleep.

Lagophthalmos: Sometimes the eye doesn't close all the way. Depends on how deep of a sleep you are referring to and if the eye has closure problems.