Intereting Posts

Gap between smooth integers tends to infinity (Stoermer-type result)?
Prove that $f(x)=0$ has no repeated roots
Where we have used the condition that $ST=TS$, i.e, commutativity?
An inequality involving Sobolev embedding with epsilon
Urysohn's Lemma needn't hold in the absence of choice. Alternate terminology for inequivalent definitions of “normal” spaces?
How to find $\omega^7$ and $\omega^6$ from $\omega^5+1=0$
How can I solve for $n$ in the equation $n \log n = C$?
A very simple discrete dynamical system with pebbles
Solve $x = \frac{1}{2}\tan(x)$
Pairwise measurable derivatives imply measurability of combined derivative
Prove inequalities $2\sqrt{n+1}-2 \leq 1 + 1/\sqrt{2} + 1/\sqrt{3} + … + 1/\sqrt{n}\leq 2\sqrt{n}-1$
Preservation of direct sums and finite generation
which of the following metric spaces are complete?
How do you find the Maximal interval of existence of a differential equation?
Difference between Analytic and Holomorphic function

Is the set of points in the plane whose coordinates are either both irrational, or both rational connected?

- Why this two spaces do not homeomorphic?
- On convergence of nets in a topological space
- Second Countability of Euclidean Spaces
- In which topological spaces is every singleton set a zero set?
- What's $\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$?
- Completion as a functor between topological rings
- Hairy Points in Infinite Graphs (and Peano Continua)
- Affine transformation
- Every subnet of $(x_d)_{d\in D}$ has a subnet which converges to $a$. Does $(x_d)_{d\in D}$ converge to $a$?
- Is there a “natural” topology on powersets?

Let $S$ be your set, and suppose $S = U \cup V$ where $U$ and $V$ are disjoint, and are both closed and open in $S$. Note that if $x$ and $y$ are both rational, then the diagonal lines $\{(x+t, y+t): t \in {\mathbb R}\}$ and $\{(x+t,y-t): t \in {\mathbb R}\}$ are subsets of $S$. Using a path of diagonal line segments, it is possible to get from any point of ${\mathbb Q} \times {\mathbb Q}$ to any other while staying in $S$. Therefore one of $U$ and $V$, let’s say $U$, contains all of ${\mathbb Q} \times {\mathbb Q}$. But ${\mathbb Q} \times {\mathbb Q}$ is dense in $S$, and $U$ is closed in $S$ so $U = S$.

For extra credit, show that $S$ is path-connected. In fact, if $a < b$ and $c < d$ with $(a,c)$ and $(c,d)$ in $S$, there is a continuous increasing function $f: [a,b] \to [c,d]$

such that $f(x)$ is rational if and only if $x$ is rational.

- Clarification of use of Cauchy-Riemann equations
- What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?
- Prove that $ k/ \langle x_1x_2,x_2x_3,x_3x_4,x_4x_1 \rangle$ is not Cohen-Macaulay.
- On an informal explanation of the tangent space to a manifold
- Area between two circles as a double integral in polar coordinates
- What does $\propto$ mean?
- ordered triplets of integer $(x,y,z)$ in $z!=x!+y!$
- A pair of continued fractions that are algebraic numbers and related to $a^2+b^2=c^m$
- Binomial coefficients: how to prove an inequality on the $p$-adic valuation?
- How this operation is called?
- Proof of Alternate, Corresponding and Co-interior Angles
- How to solve this recurrence $T(n)=2T(n/2)+n/\log n$
- Integral of Sinc Function Squared Over The Real Line
- Express $\sin^8\theta+\sin^6\theta+\sin^4\theta+\sin^2\theta-2$ as a single term in terms of $\sin\theta$
- proving that this ideal is radical or the generator is irreducible