Let y be a bijection from $(0,1)$ to $(0,1)^2$. Goal: Prove that y is not continuous.
Idea to start the proof: Makes sense assume $f$ is continuous, consider $(0,1)$ as the union of the closed intervals $[1/k, 1-1/k]$ where k goes from $1$ to $\infty$ and consider Baire's theorems:
- Suppose $C_1, C_2, \dots$ is a countable collection of closed sets in $\mathbb{R}^n$ such that the union from $n = 1$ to $\infty$ of $C_n$ contains an open set $O$. Then at least one of these closed sets contains an open set.
- Let $(M, D)$ be a complete metric space. Suppose $M$ is a countable union of closed sets of $M$. Then one of these closed sets contains an open set of $M$.
- Baire Category Theorem
Any hint it's helpful.
Hint: An injective continuous function on a compact metric space is a homeomorphism onto its image.