I need a hint: prove that [0,1] and (0,1) are not homeomorphic without referring to compactness. This is an exercise in a topology textbook, and it comes far earlier than compactness is discussed.
So far my only idea is to show that a homeomorphism would be monotonic, so it would define a poset isomorphism. But the can be no such isomorphism, because there are a minimal and a maximal elements in [0,1], but neither in (0,1). However, this doesn't seem like an elemenary proof the book must be asking for.
Answer
There is no continuous and bijective function f:(0,1)→[0,1]. In fact, if f:(0,1)→[0,1] is continuous and surjective, then f is not injective, as proved in my answer in Continuous bijection from (0,1) to [0,1]. This is a consequence of the intermediate value theorem, which is a theorem about connectedness. Are you allowed to use that?
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