a single point. Then , which represents a line-segment. Suppose and . Then , which represents a triangle. Suppose and , that is, two sets with two discrete points. then , which represents a "square". The combinatorial definition is equivalent to the topological definition in the following sense: for every two abstract simplicial complexes and , is homeomorphic to , where denotes the geometric realization of the complex . Maps Given two maps and , their join is defined based on the representation of each point in the join as , for some : Special cases The cone of a topological space , denoted , is a join of with a single point. The suspension of a topological space , denoted , is a join of with (the 0-dimensional sphere, or, the discrete space with two points). Examples 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, 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 two spheres is a sphere: the join of and is the sphere . (If and are points on the respective unit spheres and the parameter describes the location of a point on the line segment joining to , then .) The join of two pairs of isolated points is a square (without interior). The join of a square with a third pair of isolated points is an octahedron (again, without interior). In general, the join of pairs of isolated points is an -dimensional octahedral sphere. Properties Commutativity The join of two spaces is commutative up to homeomorphism, i.e. . Associativity It is not true that the join operation defined above is associative up to homeomorphism for arbitrary topological spaces. However, for locally compact Hausdorff spaces we have It is possible to define a different join operation which uses the same underlying set as but a different topology, and this operation is associative for all topological spaces. For locally compact Hausdorff spaces and , the joins and coincide. Homotopy equivalence If and are homotopy equivalent, then and are homotopy equivalent