Join

Builds on Quotient topology Product topology

Definition

Let $X$ and $Y$ be topological spaces and let $I = [0, 1]$ . The join of $X$ and $Y$ is the quotient space $X * Y = (X \times Y \times I) / \sim,$ where $X \times Y \times I$ has the product topology, and $X * Y$ carries the quotient topology. The relation $\sim$ is the equivalence relation generated by two rules. First, $(x, y, 0) \sim (x, y', 0)$ for all $x \in X$ and $y, y' \in Y$ . Second, $(x, y, 1) \sim (x', y, 1)$ for all $x, x' \in X$ and $y \in Y$ .

Remarks

The class of $(x, y, t)$ is often written $(1 - t) x + t y$ , since it lies on the segment from $x$ to $y$ . The images of $t = 0$ and $t = 1$ are copies of $X$ and of $Y$ in $X * Y$ . Example: if $Y$ is one point, then $X * Y$ is the cone on $X$ . Example: if $Y = S^0$ is two points, then $X * S^0$ is the unreduced suspension of $X$ . Example: $S^0 * S^0$ is homeomorphic to the circle $S^1$ , a square with four vertices. Non-example: the product $X \times Y \times I$ with no identification is not the join; for $X = Y = S^0$ it is four disjoint segments, while $S^0 * S^0$ is a circle.

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Product topology concept
Quotient topology concept
Join definition

Tag	`QSTN8NMSK0`
URI	`m2://QSTN8NMSK0`
Kind	definition
Writing	W0 stub
MSC	54B17, 55P99
Defines	Join
Uses	Quotient topology, Product topology
Changed	2 November 2026

Join

Definition

Remarks

Reading path

Further reading

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