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The active part of the brain controls breathing in Beluga whales and other aquatic mammals.
The ability to sleep with one eye open helps them in warding off predators.
Penguins sleep with an eye open and usually standing up.
Sleeping with one eye open allows them to not get attacked by predators.
They also make use of one eye open to be wary of bigger predators.
Yes, they do sleep with one eye shut.
Many aquatic mammals use unihemispheric sleep - in order to continue breathing while sleeping. This means dolphins literally sleep with one eye open.
Sleeping with one eye open allows these creatures to rest without getting attacked by enemies.
Making use of unihemispheric sleep, bats keep an eye out for predators.
They also use unihemispheric sleep to get some rest without drowning.