Betta fish don’t sleep, actually. They rest usually on the bottom of their fish tank. They rests in caves, or on one of his plants. He acts really startled at night and when lights are turn on. They sleep under the leaves at night, nestled in next to the stem. Sometimes they take naps during the day right in-between the leaves. Many of them will lay down in a “L-Shaped” position (head up, body to the side) and may look to be dead.
You may not have seen your fish sleep before because they usually do it at night. Assume that they are sleeping and do not really “see” you, though their eyes are wide open.
To be more precise, they rest when the lights go out. They don’t close their eyelids because they don’t have any. They float there in the open water.
It would be hard to sleep when you don’t have eyelids! Prolonged periods of lighting can also heat up the water in your tank, which is not the best for the fish.