Prove that the product spaces $[0, 1] \times (0, 1]$ and $(0, 1] \times (0, 1]$ are homeomorphic.
Show that $[0, 1)\times[0, 1)$ is homeomorphic to $[0, 1]\times[0, 1)$ but not to $[0, 1]\times[0, 1]$.