View of Montezuma's Castle, in Arizona, within the Montezuma's Castle national monument operated by the US national park service.
This is a small village built in a hole in a cliffside. Not actually a castle, and having nothing to do with Montezuma, it was built by Native Americans and later abandoned many years before Europeans arrived in the Americas.
Much later, settlers moved into the valleys nearby to start farms. When they discoverd the dwelling they thought it must have been built by the Aztecs and named it after the famous Aztec emperor.
National Park Service site on Montezuma Castle: