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In topology, the dunce hat is a compact topological 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 ...
A young boy wearing a dunce cap in class, from a staged photo c. 1906 1828 engraving showing a boy standing on a stool wearing a dunce cap with the ears of an ass. A dunce cap, also variously known as a dunce hat, dunce's cap or dunce's hat, is a pointed hat, formerly used as an article of discipline in schools in Europe and the United States—especially in the 19th and early 20th centuries ...
See Coproduct topology. Dispersion point If X is a connected space with more than one point, then a point x of X is a dispersion point if the subspace X − {x} is hereditarily disconnected (its only connected components are the one-point sets). Distance See metric space. Dowker space Dunce hat (topology)
Dunce hat (topology) Either–or topology; Excluded point topology − A topological space where the open sets are defined in terms of the exclusion of a particular ...
See Munkres Topology 2nd Edition pp. 448. Furhermore, Munkres pp. 443 gives the definition of the n-fold dunce cap as the quotient space of the unit 2-ball under the equivalence relation defined by identifying each point of the boundary (circle) with those points that are rotations of the original point by an integer multiple of 2pi/n radians.
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
In the following, it doesn't matter which definition is used.) For definiteness the reader should think of a topology as the family of open sets of a topological space, since that is the standard meaning of the word "topology". Let τ 1 and τ 2 be two topologies on a set X such that τ 1 is contained in τ 2:
The Dunce hat is contractible, but not collapsible. The cone on a Hawaiian earring is contractible (since it is a cone), but not locally contractible or even locally simply connected. All manifolds and CW complexes are locally contractible, but in general not contractible.