which of the followings are true for bijective functions which of the followings are true:-
1. There is a continuous bijection from $\mathbb{R}^2\to \mathbb{R}$.
2. There is a bijection between $\Bbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}$.

Can somebody help me please.I am totally stuck on it. 
 A: 1) Suppose such a continuous bijection $f$ exists. Let $p$ be any point in $\mathbb {R}^2$. Now consider the spaces $A=\mathbb R^2-\{p\}$ and $B=\mathbb R-\{f(p)\}$. Argue why $g:A\to B$ given by the restriction of $f$ is a continuous bijection. Now, any two points in $A$ can be joined by a curve, but not every two points in $B$ can be joined by a curve. Use that to arrive at a contradiction. 
2) Both sets are countable. This can be shown in various ways, like the general theorem that the cartesian product of any two countable sets is countable (this will help also to prove that $\mathbb Q$ is countable, solving the problem). 
A: An approach from an algebraic topologist wannabe might be the following: suppose we have a continuous bijection $\psi: \mathbb{R}^{2}\to \mathbb{R}$, then restrict it to $[0,1]^{2}$ and pick an open disc inside it (this choice is arbitrary, it's enough an open connected subset). You have, by continuity, that such a disk is sent homeomorphically (compact$\to$Hausdorff) to an open interval in $\mathbb{R}$ but this violates the Invariance of Domain.
