WASHINGTON (AP) — Smithsonian museums are closed. There are no federal staffers to answer tourists' questions at the Lincoln Memorial. And across the United States, national parks are cluttered with trash. Yet despite the federal government shutdown, a historic clock tower at the Trump International Hotel remained open Friday for its handful of visitors, staffed by green-clad National Park Service rangers.

"We're open!" one National Park Service ranger declared around lunchtime, pushing an elevator button for a lone visitor entering the site through a side entrance to ride to the top of the 31...