Join (topology)

In topology, a field of mathematics, the join of two topological spaces and , often denoted by or , is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in to every point in . The join of a space with itself is denoted by . The join is defined in slightly different ways in different contexts If and are subsets of the Euclidean space , then:,that is, the set of all line-segments between a point in and a point in . Some authors restrict the definition to subsets that are joinable: any two different line-segments, connecting a point of A to a point of B, meet in at most a common endpoint (that is, they do not intersect in their interior). Every two subsets can be made "joinable". For example, if is in and is in , then and are joinable in . The figure above shows an example for m=n=1, where and are line-segments. The join of two simplices is a simplex: the join of an n-dimensional and an m-dimensional simplex is an (m+n+1)-dimensional simplex. Some special cases are: The join of two disjoint points is an interval (m=n=0). The join of a point and an interval is a triangle (m=0, n=1). The join of two line segments is homeomorphic to a solid tetrahedron or disphenoid, illustrated in the figure above right (m=n=1). The join of a point and an (n-1)-dimensional simplex is an n-dimensional simplex. The join of a point and a polygon (or any polytope) is a pyramid, like the join of a point and square is a square pyramid. The join of a point and a cube is a cubic pyramid. The join of a point and a circle is a cone, and the join of a point and a sphere is a hypercone. If and are any topological spaces, then: where the cylinder is attached to the original spaces and along the natural projections of the faces of the cylinder: Usually it is implicitly assumed that and are non-empty, in which case the definition is often phrased a bit differently: instead of attaching the faces of the cylinder to the spaces and , these faces are simply collapsed in a way suggested by the attachment projections : we form the quotient space where the equivalence relation is generated by At the endpoints, this collapses to and to .