Prove that the product spaces $[0, 1] \times (0, 1]$ and $(0, 1] \times (0, 1]$ are homeomorphic. Let $[0, 1]$ and $(0, 1]$ be given in the standard topology.
Prove that the product spaces $[0, 1] \times (0, 1]$ and $(0, 1] \times (0, 1]$ are homeomorphic.

You could use the homeomorphism on the unit square which consists of five convex-linear maps, each one sending a triangle on the left homeomorphically to the triangle with the same number on the right.

The restriction of this map to $[0,1]\times(0,1]$ should give you the homeomorphism you want.
