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The product of a family {X_i}_(i in I) of objects of a category is an object P=product_(i in I)X_i, together with a family of morphisms {p_i:P->X_i}_(i in I) such that for ...

A compactification of a topological space X is a larger space Y containing X which is also compact. The smallest compactification is the one-point compactification. For ...

A covering map (also called a covering or projection) is a surjective open map f:X->Y that is locally a homeomorphism, meaning that each point in X has a neighborhood that is ...

Let a knot K be parameterized by a vector function v(t) with t in S^1, and let w be a fixed unit vector in R^3. Count the number of local minima of the projection function ...

A cross section of a solid is a plane figure obtained by the intersection of that solid with a plane. The cross section of an object therefore represents an infinitesimal ...

Let DeltaA^'B^'C^' be a triangle perspective to a reference triangle DeltaABC with perspector D^(''). Let A^('') be the intersection of lines BC^' and CB^', B^('') the ...

A differential k-form can be integrated on an n-dimensional manifold. The basic example is an n-form alpha in the open unit ball in R^n. Since alpha is a top-dimensional ...

The fourth group isomorphism theorem, also called the lattice group isomorphism theorem, lets G be a group and let N⊴G, where N⊴G indicates that N is a normal subgroup of G. ...

The great rhombic triacontahedron, also called the great stellated triacontahedron, is the dual of great icosidodecahedron uniform polyhedron. It is a zonohedron and a ...

A homogeneous space M is a space with a transitive group action by a Lie group. Because a transitive group action implies that there is only one group orbit, M is isomorphic ...