Not necessarily, though that may often be the case. Look through Part 97. You won't find any mention of open or closed repeaters. If one of the repeater directories has a repeater listed a certain way, that doesn't affect the control operator's right to limit the use of the repeater. There's no legal meaning to the open/closed status in a directory listing, though it may provide a hint as to the owner's attitude toward newcomers. Often, "open" means that anyone is welcome, unless and until they violate one of the rules. How they're notified of the rules and of a violation is up to the repeater owner.