Yes a landlord can look into your closet when they inspect the place because they need to look for possible mold damage or other water damage that could come in through the roof.
The landlord should not go through your clothes or belongings though but they may want you to move them around just so they can inspect the closet walls.
If you have anything in the closet that you do not want the landlord to see you should remove the items and put them in your car or somewhere else until the inspection is over with.
A landlord is only allowed to show up unannounced without advanced notice for emergencies and enter the rental unit to do emergency repairs.
However the landlord can show up without announcing it to you but they cannot enter without advanced notice unless it's an emergency.
That means that for inspections of the rental the landlord must give you 24 to 48 hour notice before they come by to enter the apartment or house to inspect.
The landlord should not go through your things and will just look over the rental unit to be sure no damage is done and to make sure everything is okay with the apartment or home.
So if you have anything that you do not want the landlord to see you should put them in your car or hide them in the closet.
The landlord might look in the closet but most will not and will just look in the rooms and take note of any possible damage etc.
A landlord inspection should take place every 6 months or once per year.