Intereting Posts

Evaluating $\int_0^\pi\arctan\bigl(\frac{\ln\sin x}{x}\bigr)\mathrm{d}x$
Finding the limit of a quotient
How should I understand the $\sigma$-algebra in Kolmogorov's zero-one law?
Does the harmonic series converge if you throw out the terms containing a $9$?
sum of the product of consecutive legendre symbols is -1
Orders, Partial Orders, Strict Partial Orders, Total Orders, Strict Total Orders, and Strict Orders
Why is full- & faithful- functor defined in terms of Set properties?
What is the simplest way to show that $\cos(r \pi)$ is irrational if $r$ is rational and $r \in (0,1/2)\setminus\{1/3\}$?
Elements as a product of unit and power of element in UFD
Functions determine geometry … Riemannian / metric geometry?
Prove that $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A + B)$
Some questions about Banach's proof of the existence of continuous nowhere differentiable functions
Dividing $n$ gon into 4 equal parts
Sum of irrational numbers, a basic algebra problem
Weak convergence in a subspace

I have the following question:

$\mathbb R^3\setminus \mathbb Q^3$ is a union of disjoint lines. This exercise is in the book: Set theory for the working mathematician of Krzysztof Ciesielski.

someone can give me a help on how to proceed?

- Subsets of the reals when the Continuum Hypothesis is assumed false
- How to prove that $\mathbb{Q}$ is not the intersection of a countable collection of open sets.
- What is the generating set of the Vietoris topology?
- Uncountable union of multiples of measurable sets.
- The collection of all compact perfect subsets is $G_\delta$ in the hyperspace of all compact subsets
- Bijection between closed uncountable subset of $\Bbb R$ and $\Bbb R$.

- $\mathbb R$ vs. $\omega+\omega$
- $\Sigma_{\alpha+1}^0(X)$ universal set based on the Baire space
- Using Axiom of Choice To Find Decreasing Sequences in Linear Orders
- Applications of ultrafilters
- Axiom of Choice and finite sets
- Every Number is Describable?
- Is a chain-complete lattice a complete lattice without the axiom of choice?
- exercise VII.G5 in kunen
- Formally proving the consistency of a formal theory
- How do we know an $ \aleph_1 $ exists at all?

Let $\mathbb{R}^3\setminus\mathbb{Q}^3=(p_\alpha)_{\alpha<\frak{c}}$. We will find lines $r_\alpha\subset\mathbb{R}^3\setminus\mathbb{Q}^3$ for all $\alpha<\frak{c}$ such that $r_\alpha\cap r_\beta=\emptyset$ for $\alpha\ne\beta$ and $p_\alpha\in\bigcup_{\beta\leq\alpha}r_\beta$ $-$ we will allow that some (but not all, of course) of the $r_\alpha$ be empty. For $r_0$ choose any line with no rational points that contains $p_0$. Now, suppose you already have defined $r_\beta$ for all $\beta<\alpha$ for some $\alpha<\frak{c}$. If $p_\alpha\in\bigcup_{\beta<\alpha}r_\beta$, then let $r_\alpha=\emptyset$. If not, notice that there exists at least one plane $\mathcal{P}$ such that $p_\alpha\in \mathcal{P}$ and $r_\beta\not\subset \mathcal{P}$ for all $\beta<\alpha$ (this can be done because there are $\frak{c}$ many planes passing through $p_\alpha$ and, in the “worst case” there are $|\alpha|<\frak{c}$ many pairwise “non-coplanar” lines). Finally, notice two things:

1) the plane $\mathcal{P}$ may intercept each $r_\beta$ at most one time;

2) there are at most countable many rational points in $\mathcal{P}$.

Since the lines in $\mathcal{P}$ that pass through $p_\alpha$ are determined by the points in $\mathcal{P}\setminus\{p_\alpha\}$, by (1) and (2) certainly there is a line $r_\alpha\subset \mathcal{P}\cap\mathbb{R}^3\setminus\mathbb{Q}^3 $ such that $p_\alpha\in r_\alpha$ and $r_\alpha\cap(\bigcup_{\beta<\alpha}r_\beta)=\emptyset$.

Thus, there exists a decomposition of $\mathbb{R}^3\setminus\mathbb{Q}^3$ in pairwise disjoint lines.

- Can I find $\sum_{k=1}^n\frac{1}{k(k+1)}$ without using telescoping series method?
- When does the kernel of a function equal the image?
- Why are removable discontinuities even discontinuities at all?
- Origin of the dot and cross product?
- Asymptotics of binomial coefficients and the entropy function
- How do I find the Intersection of two 3D triangles?
- Contractions mappings bijective maps boundarys on boundarys?
- A proof that the Cantor set is Perfect
- Tips on writing a History of Mathematics Essay
- Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$
- How to show $S^n$ is not contractible without using Homology..
- How to determine the Galois group of irreducible polynomials of degree $3,4,5$
- Distinct digits in a combination of 6 digits
- Dynkin's Theorem, and probability measure approximations
- Nonconstant polynomials do not generate maximal ideals in $\mathbb Z$