Associativity of topological join and join of spheres

If $A$ and $B$ are subsets of $\mathbb R^n$ then you can map $A\times I\times B$ to $\mathbb R^n$ by $(a,t,b)\mapsto (1-t)a+tb$. If the resulting continuous map $A*B\to \mathbb R^n$ happens to be one to one then you can ask whether it gives a homeomorphism to its image. If $A$ and $B$ are compact, then the answer is necessarily yes, since a continuous bijection from a compact space to a Hausdorff space is always a homeomorphism.

This suffices to see that the join of spheres is a sphere.

Actually there are two useful topologies on the join $X * Y$ using initial topologies, or final topologies. The former is useful for maps into the join, and the latter for maps out of the join. With initial topology, the join is associative. With final topologies, the standard problem is that the product of identification maps is not an identification map, unless one works in a convenient category of spaces. These questions are discussed in Section 5.7 of Topology and Groupoids, where associativity with the initial topology is proved. (This was in the 1968, 1988 editions!) Of course the two topologies agree on compact Hausdorff spaces, but I have not managed to write down a proof that the two topologies are equivalent in the category of compactly generated spaces. Am I missing something easy?

Later: just to add some details. One thinks of the join as lines joining points of $X$ to points of $Y$ in some Euclidean space (see Tom's answer). So one writes these elements as formal combinations $rx+sy$ where $x \in X, y \in Y$ and $0 \leqslant r,s$ and $ r+s =1$. Of course $0x+1y$ means just $y$ and $1x + 0y$ means just $x$. The initial topology is with respect to the functions $$rx+sy \mapsto r, rx+sy \mapsto s, rx+sy \mapsto x, rx+sy \mapsto y$$ which are well defined on subsets of $X*Y$. With this notation, associativity becomes easy, since a point of $X*Y*Z$ is of the form $rx+sy +tz$, with the obvious conventions and conditions; continuity of the associativity map rolls out with these initial topologies.

You may think of join $A*B$ as a sphere in the product $(\mathop{\rm Cone}A)\times (\mathop{\rm Cone}B)$.

Once you made all this precise in topological category, the associativity will follow from associativity of Cartesian product.