Join of topological spaces

From Topospaces
(Redirected from Join)
Jump to: navigation, search

Template:Product notion for topospaces

This article describes a construct that involves some variant of taking a product of a topological space with the unit interval and then making some identifications, typically at the endpoints, based on some specific maps.
View more such constructs

Definition

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

(x,y_1,0) \sim (x,y_2,0) \forall x \in X, y_1,y_2 \in Y

and

(x_1,y,1) \sim (x_2,y,1) \forall x_1,x_2 \in X, y \in Y

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

Particular cases

Cone space

Further information: Cone space

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

Suspension

Further information: suspension

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

Simplex

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

Operation properties

Template:Commutative product notion for topospaces

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

Template:Associative product notion for topospaces

There is a canonical isomorphism between (X * Y) * Z and X * (Y * Z).