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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)
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
This is a list of useful examples in general topology, a field of mathematics. Alexandrov topology; Cantor space; Co-kappa topology Cocountable topology; Cofinite topology; Compact-open topology; Compactification; Discrete topology; Double-pointed cofinite topology; Extended real number line; Finite topological space; Hawaiian earring; Hilbert cube
General topology grew out of a number of areas, most importantly the following: the detailed study of subsets of the real line (once known as the topology of point sets; this usage is now obsolete) the introduction of the manifold concept; the study of metric spaces, especially normed linear spaces, in the early days of functional analysis.
A key concept in defining simplicial homology is the notion of an orientation of a simplex. By definition, an orientation of a k-simplex is given by an ordering of the vertices, written as (v 0,...,v k), with the rule that two orderings define the same orientation if and only if they differ by an even permutation.
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.