= 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 \b[neither product is homeomorphic to the <closed> square].
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