Pen's Nest

Yeah, closet would be built in. If it's a separate unit, it'd be called a cabinet. But a medicine cabinet is like a mirror or a small box attached to the wall by the sink that opens up.

And then there's the bathroom cabinet/counter, which is a little different... there might be doors or drawers beneath the sink, and I call that a cabinet or a counter. Or I'd say like, "The toilet paper is under the sink."

Now I really miss having a medicine cabinet :P