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 . Definitions The join is defined in slightly different ways in different contexts Geometric sets 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. Topological spaces 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 . If and are bounded subsets of the Euclidean space , and and , where are disjoint subspaces of such that the dimension of their affine hull is (e.g. two non-intersecting non-parallel lines in ), then the topological definition reduces to the geometric definition, that is, the "geometric join" is homeomorphic to the "topological join": Abstract simplicial complexes If and are any abstract simplicial complexes, then their join is an abstract simplicial complex defined as follows: The vertex set is a disjoint union of and . If and are already disjoint, then one can define . Otherwise, one can define, for example, (adding 1 and 2 ensures that the elements in the union are disjoint). The simplices of are all unions of a simplex of with a simplex of . If and are disjoint, then . Examples: Suppose and , that is, two sets with