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These are a few problems on functions which I could not solve.

Which of the statements are true?

  1. There is a continuous bijection from $\mathbb{R}^2$ → $\mathbb{R}$.
  2. There is a bijection between $\mathbb{Q}$ and $\mathbb{Q} × \mathbb{Q} $.
  3. If $f : [0; 1] → [-π,π]$ is a continuous bijection then it is a homeomorphism.

Help me please. Thank you.

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1 Answer 1

Some hints:

  1. Let $\Delta$ be a triangle in $\mathbb{R}^2$. What would the image of $\Delta$ in $\mathbb{R}$ look like?
  2. $\mathbb{Q}$ is bijective with $\mathbb{N}$. Is there a bijection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$?
  3. For $f$ to be a homeomorphism and not just a continuous bijection, it needs to map open sets to open sets, or equivalently (since $f$ is a bijection), closed sets to closed sets. Can you think of a reason why $f$ maps closed (hence compact) subsets of $[0,1]$ to closed subsets of $[-\pi,\pi]$?
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