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	==Definition==		==Definition==

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{{product notion for topospaces}}

==Definition==

Given two topological spaces <math>X</math> and <math>Y</math>, the '''join''' of <math>X</math> and <math>Y</math>, denoted <math>X * Y</math>, is defined as follows: it is the quotient of the space <math>X \times Y \times I</math> under the identifications:

<math>(x,y_1,0) \sim (x,y_2,0) \forall x \in X, y_1,y_2 \in Y</math>

and

<math>(x_1,y,1) \sim (x_2,y,1) \forall x_1,x_2 \in X, y \in Y</math>

Pictorially, we can think of this as the space of all line segments joining points in <math>X</math> and <math>Y</math>, with two line segments meeting only at common endpoints.

==Particular cases==

===Cone space===

{{further|[[Cone space]]}}

The cone space of a topological space <math>X</math> can be viewed as the join of <math>X</math> with a one-point space.

===Suspension===

{{further|[[suspension]]}}

The suspension of a topological space <math>X</math> can be viewed as the join of <math>X</math> with a two-point space.

===Simplex===

The <math>n</math>-simplex can be viewed, at least topologically, as the join of <math>n</math> one-point spaces.

==Operation properties==

{{commutative product notion for topospaces}}

There is a canonical isomorphism between <math>X * Y</math> and <math>Y * X</math>, sending <math>(x,y,t)</math> to (y,x,1-t)</math>.

{{associative product notion for topospaces}}

There is a canonical isomorphism between <math>(X * Y) * Z</math> and <math>X * (Y * Z)</math>.

Join of topological spaces - Revision history

Vipul at 23:27, 9 October 2010

Vipul: 1 revision

Vipul at 13:56, 22 May 2007