The TopsI do not have any halves of hats.
I have only whole hats. I have wide hats, good for rain, and hats with simpler brims, better suited for snow. It is arguable that the armored hat for cycling might be better named a helmet. I had a cap once, but I no longer know where it is; I would not call it a hat. A cap is a separate category, and not a half a hat.
It is unclear to me, if I had had a half a hat, which half of the hat I would have. The fronts of hats do good work to keep the elements from the eyes; the backs of hats to their work to keep the same from finding an uncomfortable path behind the collar. Hats without sides do injustice on poor ears. Hats without tops are not hats.
These days, few folks wear hats.

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