Join of topological spaces

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.
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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

There is a canonical isomorphism between $X * Y$ and $Y * X$, sending $(x,y,t)$ to (y,x,1-t)[/itex].

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