Intereting Posts

Dimension of vector space of matrices with zero row and column sum.
Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$
Inequality between two sequences preserved in the limit?
Group of order 15 is abelian
Expressing $1 + \cos(x) + \cos(2x) +… + \cos(nx)$ as a sum of two terms
Definitions of Hessian in Riemannian Geometry
Irreducible polynomial means no roots?
Evaluate $\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx$
Is there a number whose absolute value is negative?
If a group mod its commutator subgroup is cyclic, then the group is abelian?
$\sigma$-algebra of well-approximated Borel sets
Completion and algebraic closure commutable
Find all Laurent series of the form…
Jacobian matrix of the Rodrigues' formula (exponential map)
How to Understand the Definition of Cardinal Exponentiation

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

- Visualizing products of $CW$ complexes
- If a product is normal, are all of its partial products also normal?
- Every compact metric space is complete
- Fundamental group of quotient of $S^1 \times $
- Topology needed for differential geometry
- How to prove this result involving the quotient maps and connectedness?
- Open Dense Subset of $M_n(\mathbb{R})$
- When is the image of a null set also null?
- How many metrics are there on a set up to topological equivalence?
- Borel Measures: Atoms vs. Point Masses

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.

- Why is “for all $x\in\varnothing$, $P(x)$” true, but “there exists $x\in\varnothing$ such that $P(x)$” false?
- Stochastic processes question – random walk hitting time
- On Constructions by Marked Straightedge and Compass
- $\{m \alpha, m \in \mathbb Z\}$is dense in $$ for $\alpha$ irrational
- How to prove that given set is a connected subset of the space of matrices?
- To estimate $\sum_{m=1}^n \Big(d\big(m^2\big)\Big)^2$
- Vector Algebra Coordinate Transformation
- Show that if $f(x) > 0$ for all $x \in $, then $\int_{a}^b f(x) dx > 0$
- difference between a $G$ invariant measure on $G/H$ and a haar measure on $G/H$
- Can the “radius of analyticity” of a smooth real function be smaller than the radius of convergence of its Taylor series without being zero?
- Picking random points in the volume of sphere with uniform probability
- There are apparently $3072$ ways to draw this flower. But why?
- Does every open manifold admit a function without critical point?
- Prove that:$\int_{0}^{1}{x+x^2+\cdots+x^{2n}-2nx\over (1+x)\ln{x}}dx=\ln{\left}$
- Minkowski sum of two disks