Past exam of the topology course of the University of Oxford / 2014 / 2 / b / i / Solution

The interval $[0,1]$ is compact by the Heine-Borel theorem, so its square is compact by part (a). The half-open interval $[0,1)$ is not compact because it is not closed in the real line.

Each of the other two products has a continuous surjective coordinate projection onto $[0,1)$. If either product were compact, that image would be compact, a contradiction. Compactness is preserved by homeomorphisms, so neither product is homeomorphic to the closed square.