I think the following is a good exercise for first course in real analysis.
Does there exist continuous bijection from to ? (1)
Well, it feels like the answer should be “no”. At least if the question was
Does there exist continuous bijection from to ? (2)
the answer would be a trivial “no”. Continuous image of compact set is compact; in particular, is compact, and is not compact, so continuous bijection from to cannot exist. So it is natural to wonder how questions (1) and (2) are related. If is a continuous bijection, then is well-defined function. But, if in addition, is known to be continuous, then once again we have a continuous bijection from which we know is impossible. For general reference:
Definition. When is a continuous bijection, and is continuous, then we say that is homeomorphism from to .
Thus, being homeomorphic is a stronger property than being continuous and bijective. Roughly speaking, homeomorphisms preserve all intrinsic topological properties (e.g. compactness), just like how group homomorphisms preserve the group structure. To naively answer question (1), it would be very good to know when continuous bijections are homeomorphisms. Because if continuous bijections were always homeomorphisms, then the answer to question (1) is definitive “no” by what we explained above. Unfortunately, that’s not always the case. But not all is lost. We have the following sufficient condition:
Theorem. If is continuous bijection, where is compact and is Hausdorff, then is a homeomorphism.
But the domain of is , so we can’t apply the above theorem. What a pity! After much teasing, I think it would only be fair to present the complete answer:
Solution to Question (1). Suppose that is a continuous bijection. Then there exists a unique such that . Let be so small that . So the intervals and get mapped under to intervals and , respectively for some . But then every value in would be achieved at least twice by , contradiction.
Acknowledgement. I learnt the above proof from t.b.’s answer here:
Other proofs given in that thread are also very nice.