The toilet lid is hinged to open and close over the toilet bowl. The lid is closed when the toilet is not in use. You should shut the toilet lid for hygiene and safety reasons.
The toilet lid fits over the toilet seat and protects the interior toilet.
According to the Bog Standard, a British campaign to improve school toilets, if you do not shut the toilet lid before flushing, flushing throws fine water spray outside the toilet. (Reference 1)
Toilet spray due to an open toilet lid spreads viruses and bacteria from the toilet onto the outside of the toilet and into the room. (Reference 1)
Keep the lid closed so that objects like bathroom hair dryers or other products do not fall in the water.
Shut the lid to keep children or animals from getting into the toilet and drinking dirty water or drowning.
According to Seattle Public Health, sewer rats follow food up the pipes to toilets. Keep your toilet lid down. Flush a rat away by squirting in a little dishwashing liquid and then flushing.