Intereting Posts

Given $a,b$, what is the maximum number which can not be formed using $na + mb$?
Do all Lie groups admit deformation retracts onto compact subgroups?
Choosing $\lambda$ to yield sparse solution
Isomorphism between $I_G/I_G^2$ and $G/G'$
Proving $n^{97}\equiv n\text{ mod }4501770$
Limits in the category of exact sequences
Evaluate $\displaystyle\int_2^3 {\text{d}x\over x \log(x + 5)}$
Comparison theorem for systems of ODE
Why is $n$ divided by $n$ equal to $1$?
How to rotate n individuals at a dinner party so that every guest meets every other guests
linear algebra over a division ring vs. over a field
Prove any function $f$ is Riemann integrable if it is bounded and continuous except finite number of points
${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$
$2n^2-\lfloor m^b\rfloor=k$ has only finitely many integer solutions
If any x points are elected out of a unit square, then some two of them are no farther than how many units apart?

There are a lot of websites and forums, which explain that there is a bijection between $\mathbb{R}$ and $\mathbb{R}^2$, and even give some bijections. (By the way: Can you generalize it? since it works with the natural Numbers and the real numbers, does there exist a bijection between any infinite set $X$ and $X^2$) On some websites there is claimed that there doesn’t exist a continuous bijection. But how would I approach such a proof? (I don’t need a complete proof,… just a starting point because I have absolutely no idea how to begin this.)

- Infinite partition of $\mathbb N$ by infinite subsets
- substituting a variable in a formula (in logic)
- The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.
- Is empty set a proper subset of itself?
- For what conditions on sets $A$ and $B$ the statement $A - B = B - A$ holds?
- Is a continuous function on a bounded set bounded itself?
- How to solve probability when sample space is infinite?
- The largest number system
- Cardinality of the Cartesian Product of Two Equinumerous Infinite Sets
- Double Complement of a set proof

Easy direction: there is no continuous bijection from $\mathbb R^2$ onto $\mathbb R$. For suppose $f : \mathbb R^2 \to \mathbb R$ is continuous. Choose one point $a \in \mathbb R^2$. The deleted set $\mathbb R^2 \setminus \{a\}$ is connected. A continuous image of a connected set is connected. So $f\big(\mathbb R^2 \setminus \{a\}\big)$ is a connected subset of $\mathbb R$. In particular, it is not the deleted set $\mathbb R \setminus \{f(a)\}$, since that is disconnected. So either $f$ is not injective or not surjective.

**Note** The same argument shows there is no continuous bijection from an open ball in $\mathbb R^2$ onto a subset of $\mathbb R$. You just have to make sure to choose $a$ that does not map to the maximum or minimum of the image.

Hard direction: there is no continuous bijection from $\mathbb R$ onto $\mathbb R^2$. This will use the Baire Category Theorem. Let $f : \mathbb R \to \mathbb R^2$ be continuous and injective. $\mathbb R$ is the countable union of compact sets $[-n,n]$. I will show that $f\big([-n,n]\big)$ has empty interior. Then $f(\mathbb R)$ is a countable union of closed sets with empty interior, so by the Baire Category Theorem, it is not $\mathbb R^2$.

So, why does $K_n:=f\big([-n,n]\big)$ have empty interior? Suppose it has nonempty interior. Since $[-n,n]$ is compact and $\mathbb R^2$ is Hausdorff, the continuous bijection $f$ from $[-n,n]$ onto $K_n$ is a homeomorphism. So the restriction of $f^{-1}$ to an open ball contained in $K_n$ would be a homeomorphism from an open ball in $\mathbb R^2$ onto a subset of $\mathbb R$. Contradiction.

For the question about generalization, consider Netto’s theorem;

If $f$ represents a bijective map from an $m$-dimensional smooth manifold $\mu_m$ onto an $n$-dimensional smooth manifold $\mu_n$ and $m \neq n$, then $f$ is necessarily discontinuous.

- How did Archimedes find the surface area of a sphere?
- Convergence of Riemann sums for improper integrals
- regularization of sum $n \ln(n)$
- Number of unique cubes with one red cube in every $1*1*4$ segment
- Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples
- What are some martingales for asymmetric random walks?
- Morphism between projective schemes induced by injection of graded rings
- Is it possible to gain intuition into these trig compound angle formulas – and in general, final year high school math?
- Summation of Fibonacci numbers $F_n$ with $n$ odd vs. even
- An open interval is an open set?
- Combinatorial proofs of the following identities
- Number of permutations that strictly contain two consecutive vowels
- Variation of the Kempner series – convergence of series $\sum\frac{1}{n}$ where $9$ is not a digit of $1/n$.
- If the positive series $\sum a_n$ diverges and $s_n=\sum\limits_{k\leqslant n}a_k$ then $\sum \frac{a_n}{s_n}$ diverges as well
- Subgroups of symmetric group