open hat
Any headwear that atypically leaves the crown of the head exposed, either by design or due to damage.
Note that this excludes headwear that already has applicable terminology on Danbooru - such as visor caps being the 'open' variant of baseball caps.
One example of an open hat is the traditional Chinese "Cooling hat"/"Hakka hat" (Liangmao). This bears resemblance to ajirogasa, with the exception of a circular cutout in its middle. This modification is applied to keep the wearer's head cool in hot weather.
Other terms: capless hat, half-hat, topless hat