When pillbox hats were in fashion, women wound a folded band of fabric around the brim, either to add a touch of decoration or to hold their hat veil on. This band of fabric was called a çember or çenber.
Sometimes the band was embroidered instead of folded. There are several examples of these bands in the Topkapi Museum, all attributed to Hürrem Sultan. They’re labeled “kas basti,” a word that I haven’t been able to track down to any other period source but which has been picked up by the scholarly community.