What is the topological space obtained by cutting $M\times N$ along a copy of $M$ or $N$ (both closed topological spaces). E.g. if we cut a torus $\Bbb S^1\times \Bbb S^1$ along a circle then the result is a cylinder.
If the answer depends on $M$ and $N$, the simple case $\Bbb S^n\times \Bbb S^m$ suffices.
Removing a copy of $M$ from the product corresponds entirely to removing a point from $N$ before taking the product: $$ (M\times N)\setminus(M\times \{p\})=M\times(N\setminus\{p\}) $$ In the case of $S^1\times S^1$, removing a point from a circle gives an interval, and a circle times an interval yields your cylinder.
Some times the result will depend on exactly which point you remove from $N$, of course. For instance, if you have a closed cylinder shell $S^1\times [0,1]$, then removing an end point from $[0,1]$ results in a half-open cylinder, while removing an interior point results in two half-open cylinders.