The first two topologies are not comparable. Indeed, $[0,1]\times(0.5,1]$ open in the first topology contains the point $(0,1)$ , but does not contain any its open neighborhood in the second topology, and $\{0\}\times(0,1)$ open in the second topology contains the point $(0,0.5)$ , but does not contain any its open neighborhood in the first topology.

The third topology is strictly finer than the first and second one. Indeed, according to Exercise 9, the third topology is generated by the basis consisting of the sets $\{x\}\times((a,b)\cap[0,1])$ . So, every basis element $((a,b)\cap[0,1])\times((c,d)\cap[0,1])$ of the first topology is the union of some basis sets of the third topology, and every basis set $(a,b)<(x,y)<(c,d)$ of the second topology is the union of some basis sets of the third topology as well. The fact that the third topology is strictly finer than the first and second topologies follows from the fact that the first and second topologies are not comparable.