Definition:Join (Topology)

Definition

Let $X, Y$ be topological spaces.

Then, the join of $X$ and $Y$ is defined and denoted as:

$X \ast Y = \paren {X \times Y \times \closedint 0 1} / \RR$

where:

$X \times Y \times \closedint 0 1$ denotes the product space

$S / \RR$ denotes the quotient space

the equivalence relation $\RR$ is induced by the mapping:

$R : X \times Y \times \closedint 0 1 \to X \cup X \times Y \times \openint 0 1 \cup Y$

defined as:

$\map R {x, y, t} = \begin{cases} x & : t = 0 \\ \tuple {x, y, t} & : t \in \openint 0 1 \\ y & : t = 1 \end{cases}$

Sources

• 2001: Allen Hatcher: Algebraic Topology