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Geometric join of two line segments. The original spaces are shown in green and blue. The join is a three-dimensional solid in gray.

In topology, a field of mathematics, the join of two topological spacesA and B, often denoted by AB{displaystyle Aast B} or AB{displaystyle Astar B}, is defined to be the quotient space

(A×B×I)/R,{displaystyle (Atimes Btimes I)/R,}

where I is the interval [0, 1] and R is the equivalence relation generated by

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Join.nearpod.join

Join
(a,b1,0)(a,b2,0)for all aA and b1,b2B,{displaystyle (a,b_{1},0)sim (a,b_{2},0)quad {mbox{for all }}ain A{mbox{ and }}b_{1},b_{2}in B,}
(a1,b,1)(a2,b,1)for all a1,a2A and bB.{displaystyle (a_{1},b,1)sim (a_{2},b,1)quad {mbox{for all }}a_{1},a_{2}in A{mbox{ and }}bin B.}

At the endpoints, this collapses A×B×{0}{displaystyle Atimes Btimes {0}} to A{displaystyle A} and A×B×{1}{displaystyle Atimes Btimes {1}} to B{displaystyle B}.

Intuitively, AB{displaystyle Astar B} is formed by taking the disjoint union of the two spaces and attaching line segments joining every point in A to every point in B.

Examples[edit]

Joint pain
  • The join of a space X with a one-point space is called the coneCX of X.
  • The join of a space X with S0{displaystyle S^{0}} (the 0-dimensional sphere, or, the discrete space with two points) is called the suspensionSX{displaystyle SX} of X.
  • The join of the spheres Sn{displaystyle S^{n}} and Sm{displaystyle S^{m}} is the sphere Sn+m+1{displaystyle S^{n+m+1}}.
  • 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 n+1 pairs of isolated points is an n-dimensional octahedral sphere.
  • The join of two abstract simplicial complexes X and Y on disjoint vertex sets is the abstract simplicial complex {xyxX,yY}{displaystyle {xcup y xin X,yin Y}}. I.e., any simplex in the join is the union of a simplex from X and a simplex from Y. For example, if each of X and Y contain two isolated points, X = { {1}, {2} } and Y = { {3}, {4} }, then X * Y = { {1,3} , {1,4} , {2,3} , {2,4} } = a 'square' graph.

Properties[edit]

  • The join of two spaces is homeomorphic to a sum of cartesian products of cones over the spaces and the spaces themselves, where the sum is taken over the cartesian product of the spaces:
ABC(A)×BA×BC(B)×A.{displaystyle Astar Bcong C(A)times Bcup _{Atimes B}C(B)times A.}
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  • Given basepointed CW complexes (A,a0) and (B,b0), the 'reduced join'
ABA{b0}{a0}B{displaystyle {frac {Astar B}{Astar {b_{0}}cup {a_{0}}star B}}}

is homeomorphic to the reduced suspension

Σ(AB){displaystyle Sigma (Awedge B)}

of the smash product. Consequently, since A{b0}{a0}B{displaystyle {Astar {b_{0}}cup {a_{0}}star B}} is contractible, there is a homotopy equivalence Mmc's music man.

ABΣ(AB).{displaystyle Astar Bsimeq Sigma (Awedge B).}

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See also[edit]

References[edit]

  • Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN0-521-79160-X and ISBN0-521-79540-0
  • This article incorporates material from Join on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
  • Brown, Ronald, Topology and Groupoids Section 5.7 Joins.

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