---
tag: ZEETZHV69Y
kind: concept
name: Join
msc: [54B17, 55P99]
sources:
  - url: https://en.wikipedia.org/wiki/Join_(topology)
same_as:
  wikidata: Q3183033
  wikipedia: Join (topology)
---

## Summary

The join $X * Y$ of two topological spaces is the space of all segments that run from a point of $X$ to a point of $Y$.
Formally it is $X \times Y \times [0, 1]$ with the end $t = 0$ collapsed onto $X$ and the end $t = 1$ collapsed onto $Y$.
Joins generalise cones and suspensions, and the join of two spheres is again a sphere: $S^m * S^n \cong S^{m+n+1}$.

## Explanations

### L1 human

Place $X$ and $Y$ in space far from each other and draw a straight segment from every point of $X$ to every point of $Y$.
The union of all these segments is the join.
For example, the join of two points is a segment, and the join of a point with a circle is a cone.
The join of two disjoint segments in general position in $\R^3$ is a solid tetrahedron.