@Nancy Hart - aah that reminds me of a homeless gal I came to know many years ago, she was probably in her late 50s and she wore at least 4 hats on her head - she was a character ...
Ida Kromer stitched together her first ear-banded, 6-panel cap in 1903, the same year the Wright Brothers filed for a patent regarding a “flying machine.” It was a good year for innovation. Read the full Stormy Kromer history here. George “Stormy” Kromer "After repeatedly losing his cap on the wind-whipped locomotive where he works, George Kromer asks his wife, Ida, to stitch a new-fashioned baseball cap with a higher crown, a pull-down earband to keep it snug, and a soft, cloth visor. Quickly, Stormy’s fellow engineers beg him for their own six-panel caps. Ida’s sewing machine begins to sing."
You are probably right, @Hal Pollner. A different scene from the movie does NOT show any center crease in Bergman's hat. It looks a little like a pork pie with a wide brim. You might find this interesting. Hello Fedora "Fedora is a hat made out of felt. It has a wide brim, crown that is indented and pitched, and a ribbon. ... It first appeared in 1882 as a female hat. That year was a first production of a play “Fédora” by the French author Victorien Sardou. He wrote the part of Princess Fédora Romanoff, a title role, for then famous actress Sarah Bernhardt. In it she wore center-creased, soft brimmed hat. The hat was soon a popular fashion for women especially for women’s-rights activists."