In the proof of Van Kampen's theorem in Hatcher's book, which is theorem 1.20 on p43 , we read in its proof on p45 (see here: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf)
If anybody wants, I can make a screenshot and post it here.
We may assume the $s$-partition subdivides the partitions giving the
products $f_1* \dots * f_k$ and $f_1'* \dots * f_l'$.
What exactly does this line mean? What partitions giving the products $f_1*\dots *f_k$ is the author talking about? And why can we assume this?
Best Answer
The concatenation $f_1 \cdot \ldots \cdot f_k$ is a map $I \to X$ obtained by dividing the interval into $k$ subintervals and doing $f_i$ on the $i$th subinterval. You can take these subintervals to be $[0, 1/k], [1/k, 2/k], \ldots, [(k-1)/k, 1]$, or you can take the product two loops at a time; for instance, $f_1 \cdot (f_2 \cdot (f_3 \cdot f_4)))$ would use the partition $[0, 1/2]$, $[1/2, 3/4]$, $[3/4, 7/8]$, $[7/8, 1]$. This is the partition giving the product $f_1 \cdot \ldots \cdot f_k$ that Hatcher refers to (and similarly for $f_1' \cdot \ldots \cdot f_l'$).
When Hatcher says that the $s$-partition on page 45 can be assumed to subdivide the partitions giving the products, he means that the partition points (e.g. the 0, 1/2, 3/4, 7/8, and 1 in my example) each appear as one of the values $s_0, s_1, \ldots s_m$. We can assume this because if our $s$-partition doesn't contain these points, we simply refine the $s$-partition by inserting them; any refinement of the $s$-partition will still have the desired property that the rectangles $R_{ij}$ each map into a single $A_{\alpha}$.