An attachment 10 being made with a springy, yet relatively rigid material includes at least one U-shaped arm 12 which is in contact with an internal surface of a crown 22 of a headgear 20. The crown has an opening which is larger than its ordinary size for a wearer's head 30. The arm has two end portions oriented toward the left and right portions of a lower periphery of the crown respectively, spaced apart to each other in a distance shorter than a width between the left and right sides of the head, adapted to clamp the left and right sides of the head in the springy manner, also adapted to make the opening of the crown narrowed in its left and right directions while the opening is lengthened elliptically in its front and rear directions to provide air space to the head, whereby the air space can engage to ventilate scalp 32 of the head keeping cool and dry and also to prevent hair 34 of the head from being disturbed and pressed.