Product of Closed and Half-Open Unit Intervals is Homeomorphic to Product of Half-Open Unit Intervals

Theorem
Let $\closedint 0 1$ denote the closed unit interval $\set {x \in \R: 0 \le x \le 1}$.

Let $\hointr 0 1$ denote the half-open unit interval $\set {x \in \R: 0 \le x < 1}$.

Let both $\closedint 0 1$ and $\hointr 0 1$ have the Euclidean topology.

Then the product space:
 * $\closedint 0 1 \times \hointr 0 1$

is homeomorphic to:
 * $\hointr 0 1 \times \hointr 0 1$