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| [[Image:Join.svg|right|thumb|Geometric join of two [[line segment]]s. The original spaces are shown in green and blue. The join is a three-dimensional solid in gray.]]
| | Greetings! I am Marvella and I really feel comfy when people use the complete name. Puerto Rico is where he's been living for years and he will never move. It's not a typical thing but what she likes doing is base leaping and now she is attempting to earn cash with it. He utilized to be unemployed but now he is a pc operator but his marketing never comes.<br><br>my website - [http://nuvem.tk/altergalactica/NumbersdbEstepvu nuvem.tk] |
| In [[topology]], a field of [[mathematics]], the '''join''' of two [[topological space]]s ''A'' and ''B'', often denoted by <math>A\star B</math>, is defined to be the [[quotient space]]
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| :<math> (A \times B \times I) / R, \, </math>
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| where ''I'' is the [[interval (mathematics)|interval]] [0, 1] and ''R'' is the [[equivalence relation]] generated by
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| :<math> (a, b_1, 0) \sim (a, b_2, 0) \quad\mbox{for all } a \in A \mbox{ and } b_1,b_2 \in B,</math>
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| :<math> (a_1, b, 1) \sim (a_2, b, 1) \quad\mbox{for all } a_1,a_2 \in A \mbox{ and } b \in B.</math>
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| At the endpoints, this collapses <math>A\times B\times \{0\}</math> to <math>A</math> and <math>A\times B\times \{1\}</math> to <math>B</math>.
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| Intuitively, <math>A\star B</math> is formed by taking the [[disjoint union (topology)|disjoint union]] of the two spaces and attaching a line segment joining every point in ''A'' to every point in ''B''.
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| ==Properties==
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| * The join is homeomorphic to sum of cartesian products of cones over spaces and spaces itself, where sum is taken over [[cartesian product]] of spaces:
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| :<math>A\star B\cong C(A)\times B\cup_{A\times B} C(B)\times A</math>
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| and is [[homotopy equivalent]] to suspension of [[smash product]] of spaces:
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| :<math>A\star B\simeq \Sigma(A\wedge B)</math>
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| ==Examples==
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| * The join of subsets of ''n''-dimensional Euclidean space ''A'' and ''B'' is [[homotopy equivalent]] to the space of paths in ''n''-dimensional [[Euclidean space]], beginning in ''A'' and ending in ''B''.
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| * The join of a space ''X'' with a one-point space is called the [[cone (topology)|cone]] ''CX'' of ''X''.
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| * The join of a space ''X'' with <math>S^0</math> (the 0-dimensional [[sphere]], or, the [[discrete space]] with two points) is called the [[Suspension (topology)|suspension]] <math>SX</math> of ''X''.
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| * The join of the spheres <math>S^n</math> and <math>S^m</math> is the sphere <math>S^{n+m+1}</math>.
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| ==See also==
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| *[[Cone (topology)]]
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| *[[Suspension (topology)]]
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| ==References==
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| *[[Allen Hatcher|Hatcher, Allen]], [http://www.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic topology.''] Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
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| *{{PlanetMath attribution|id=3985|title=Join}}
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| [[Category:Algebraic topology]]
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| [[Category:Binary operations]]
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Greetings! I am Marvella and I really feel comfy when people use the complete name. Puerto Rico is where he's been living for years and he will never move. It's not a typical thing but what she likes doing is base leaping and now she is attempting to earn cash with it. He utilized to be unemployed but now he is a pc operator but his marketing never comes.
my website - nuvem.tk