Let $X$ and $Y$ be topological spaces and let I be the unit interval. The join $X*Y$ is defined as $$(X\times Y\times I)/\sim\,\,\mbox{with }\,\,I = [0; 1]$$ and the relation is given by $(x,y,0)\sim (x′,y,0), (x,y,1)\sim (x,y′,1)$ for $x\in X, y\in Y$.
I don't understand $\sim$ relation, i.e, $$\forall (x; y; t),\,\,(x'; y'; t')\in X\times Y\times I,\,\, (x; y; t) \sim (x'; y'; t') \leftrightarrow ??$$
Can you explain for me? Thank you very much!
On
You need to go back to the intuitive idea of the join.
Suppose $X,Y$ are two subsets of Euclidean space of sufficiently high dimensions that the line segments joining points $x \in X$ to points $y \in Y$ do not meet. The union of these line segments is then the join $X * Y$. Its points are $rx + sy$ where $r,s\geqslant 0$, and $r+s=1$. Here are some pictures
$$\text{Fig.5.5}$$
taken from Topology and Groupoids: pdf, Section 5.7.
Of course we do not want to use generally an embedding in Euclidean space, so we define $X*Y$ to consist of "formal points" $rx + sy$ with $r,s$ as above except that if $r=0$ we ignore $rx$, getting just $y,$ and if $s=0$ we ignore $sy$, getting just $x$. A useful topology on $X*Y$ is then an initial topology as detailed in that Section, and which I won't labour on here. This topology makes the join associative.
However a more commonly used topology is a final topology as an identification space of $X \times Y \times [0,1]$.
The initial topology is good for deciding continuity of maps into the join, while the final topology is good for deciding continuity of maps out of the join.
The relation says $(x, y, t) \sim (x', y', t')$ if and only if either 1) $y = y'$ and $t = t' = 0$ or 2) $x = x'$ and $t = t' = 1$