Will the mail be delivered? Would post offices be open?
According to the House Committee on Oversight and Government Reform, the U.S. Postal Service would continue mail delivery, retail service, and other operations in the event of a government shutdown. The Postal Service is essentially funded through the sale of postage.
Are National Parks and Museums (Smithsonian) shut down?
According to the House Committee on Natural Resources, parks would be closed to public use. “Critical” personnel would be kept in place for resource protection (and to tell people that arrive that they are closed). The National Park Service has informed the Natural Resources Committee that they do not expect to close access to open-entrance park land, but everywhere they can lock a gate, close a road, and shutoff amenities, it will be done. Although the Washington Monument is already closed for repairs, we understand that the NPS will attempt to obstruct public access to other popular monuments on the National Mall with closures and barricades. According to CRS Report RL34680, while not indicative of future behavior, 368 National Park Service sites (loss of 7 million visitors) reportedly occurred in a previous shutdown.
