How public is public space?

Public space is supposed to be accessible and open for all people. It contains parks and public libraries etc. The spaces in New York that are public space are problematic. Either they are closed off and barricaded for fear of terrorism or other problems exist. For example: New York University rents out Washington Square Park every year for graduation. During the week of graduation, the park is closed off to basically anyone who cannot produce an NYU identification card. Also, I was thrown out Washington Square Park twice. Once, because Will Smith rented the park to film a movie. Second, because our event "Rock da mic" supposedly violated the noise ordinance. The parks in New York also close at weird times. For the city that never sleeps, you have to leave Christopher Street Pier by midnight unless you would like a police escort. How public is public space if I can't rap on the mic,and can't walk through the park at any time?