How can we construct a $1-1$ correspondence between $(0,1$) and $\mathbb{R}$ that transform irrationals into irrationals and rationals into rationals?
1117
Is there a bijection between $(0,1)$ and $\mathbb{R}$ that preserves rationality?
41767
Sequence of rational numbers converging to an arbitrary irrational number $\beta$.
41872
Show that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^2 \setminus\{(0,0)\}$
93017
Isomorphism between $\Bbb R$ and $\Bbb R(X)$?
10008
Proof that the nth root of a rational s/t is irrational unless s and t are perfect nth powers
72642
Continuous bijection from $(a,b) \to S^1$?
136538
How can I show that a bijection between $[0,1)$ and $(0,1)$ cannot be continuous?
6227
Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$
31244
Is this a correct bijection between $(0,1]$ and $[0,1]$?
106884
Is there a continuous bijection between an interval $[0,1]$ and a square: $[0,1] \times [0,1]$?
142000
Prove that there is an irrational number and a rational number between any two distinct real numbers
87735
Prove that the roots $\in \Bbb R$ of $x^3+x+1=0$ aren't rational without RRT
27670
Group homomorphisms $\mathbb{Q}\to\mathbb{R}$
31710
Show that $\mathcal{J}=\{[a, a+1) \mid a \in \mathbb{R}\}$ generates the Borel set of $\mathbb{R}$
121685
Is there a bijection of the natural numbers which swaps $\frac{1}{n}$-summable subsets with $\frac{1}{\sqrt{n}}$-summable subsets?
23752
Proving Irrationality
149605
Homeomorphism from $[0,1]\times[0,1]$ to $\overline{D}(0,1)$?