Wear a hat or a cap in cold weather?

Hat, velvet hat used in winter, should use this word; In summer, there are eaves around the cool hat to use this word; Top hat uses this word.

Cap, usually refers to hats, such as baseball caps, use this word; A hat worn by a policeman or soldier with a brim in front, use this word.

This is the difference. So, I think you should wear a hat in winter.