If your pet is acting like he's asleep but his eyes are open, don't be alarmed -- humans close their eyes when they sleep, but not all dogs do. It depends partially on the breed and partially on the individual dog, but your dog may be capable of keeping his eyes open, even just a little, while he's asleep.
Open and Closed
When a dog sleeps with his eyelids open, the eyes may roll back in his head, leaving just the whites exposed. Whether they're closed or not, if you look closely, you may even see the eyes moving. This is a natural part of REM sleep that dogs go through, just like humans do. You shouldn't necessarily expect your dog to sleep with his eyes open, though, because certain breeds are simply more predisposed to it than others. Greyhounds, for example, or dogs with bulbous eyes, like pugs, are likeliest to show at least a little eye while they're asleep.