# Reduced suspension

From Topospaces

Template:Self-functor on topospaces

## Definition

Given a topological space with basepoint , the **reduced suspension** of is defined in the following equivalent ways:

- It is the smash product of with (we could choose to be any point in (it does not matter since is a homogeneous space).
- It is the quotient of the suspension of by the identification:

The reduced suspension of is denoted by .

## Facts

The reduced suspension commutes with joins. In other words: