# Join of topological spaces

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.

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## Contents

## Definition

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

and

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

## Particular cases

### Cone space

`Further information: Cone space`

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

### Suspension

`Further information: suspension`

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

### Simplex

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

## Operation properties

Template:Commutative product notion for topospaces

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

Template:Associative product notion for topospaces

There is a canonical isomorphism between and .