Файл:DunceHatSpace.pngTo get a dunce hat, take a solid triangle and successively glue together all three sides with the indicated orientation.
In topology, the dunce hat is a compacttopological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed. Simply gluing two sides oriented in the opposite direction would yield a cone much like the dunce cap, but the gluing of the third side results in identifying the base of the cap with a line joining the base to the point.
The dunce hat is contractible, but not collapsible. Contractibility can be easily seen by noting that the dunce hat embeds in the 3-ball and the 3-ball deformation retracts onto the dunce hat. Alternatively, note that the dunce hat is the CW-complex obtained by gluing the boundary of a 2-cell onto the circle. The gluing map is homotopic to the identity map on the circle and so the complex is homotopy equivalent to the disc. By contrast, it is not collapsible because it does not have a free face.
Файл:Dunce hat animated.gifDunce hat Folding. The blue hole is only for better view: it may be filled by a spherical cap. The (green) triangle border folds on a circle.