Intereting Posts

Finding the largest subset of factors of a number whose product is the number itself
Calculating in closed form $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^4(m^2+n^2)}$
Cover of (0,1) with no finite subcover & Open sets of compact function spaces
Limit of $L^p$ norm
The notion of complex numbers
Showing that the map $f(z) = \frac{1}{z} $ maps circles into circles or lines
An application of partitions of unity: integrating over open sets.
Why is $ \sum_{n=0}^{\infty}\frac{x^n}{n!} = e^x$?
Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem.
A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I
In category theory: What is a fibration and why should I care about them?
How to determine the equation and length of this curve consistently formed by the intersection of Circles
The chain rule, how to interpret it
Is Tolkien's Middle Earth flat?
Eulers totient function divided by $n$, counting numbers in the set that are coprime to n

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.)

- Is the collection of finite subsets of $\mathbb{Z}$ countable?
- Does any uncountable set contain two disjoint uncountable sets?
- Which axiom shows that a class of all countable ordinals is a set?
- Proving Separation from Replacement
- Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
- Finite Cartesian Product of Countable sets is countable?
- When do two functions become equal?
- The set of real numbers and power set of the natural numbers
- Uniform Continuity implies Continuity
- Is this a Correct Proof of the Principle of Complete Induction for Natural Numbers in ZF?

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.

- Mapping homotopic to the identity map has a fixed point
- Lagrange polynomial and derivative problem
- “Too simple to be true”
- Variety of Nilpotent Matrices
- Fourier transform as diagonalization of convolution
- generalization of geometric series $ \sum_{k=0}^\infty x^{p(k)} $
- Exotic maps $S_5\to S_6$
- How can I prove that a matrix is area-preserving?
- Determining similarity between paths (sets of ordered coordinates).
- Using functions to separate a compact set from a closed set in a completely regular space
- Correct notation for “slice” of Integers
- Projective resolution of $k$ over $R=k/(xy)$
- First Course in Linear Algebra book suggestions?
- An identity wich applies to all of the natural numbers
- Looking for a source of an infinite trigonometric summation and other such examples.