Intereting Posts

Greatest prime factor of $n$ is less than square root of $n$, proof
Killing vector fields restricted to geodesics
$\Omega=x\;dy \wedge dz+y\;dz\wedge dx+z\;dx \wedge dy$ is never zero when restricted to $\mathbb{S^2}$
A Schwartz function problem
The only 1-manifolds are $\mathbb R$ and $S^1$
Why is the fundamental group a sheaf in the etale topology?
Continuous bijection from $(0,1)$ to $$
Finding the volume of two intersecting cylinders at arbitrary angles
asymptotic expansion from 3 leading terms
Pullbacks of categories
Difference Between Tensor and Tensor field?
Real valued analytic function defined on a connected set is constant
A decomposition of a differentiable function
Why can't a set have two elements of the same value?
Evaluate definite integral $\int_{-1}^1 \exp(1/(x^2-1)) \, dx$

Is there a bijective continuous function $f:\mathbb{Q}\rightarrow \mathbb{Q}$ that not a homeomorphism?

I am not able to prove it or disprove it. The problem that the rationals is not even locally compact.

- Prove that a continuous image of a closed subset of a compact space is a closed subset
- Determine the closure of the set $K=\{\frac{1}{n}\mid n\in\mathbb N\}$ under each of topologies
- Example to show the distance between two closed sets can be 0 even if the two sets are disjoint
- Origins of the modern definition of topology
- Prove that the complement of $\mathbb{Q} \times \mathbb{Q}$. in the plane $\mathbb{R}^2$ is connected.
- On convergence of nets in a topological space

- formalize definition of topology
- Topologically equivalent metric
- Compact sets in regular topological spaces
- Question asked on the structure of open sets in $\mathbb{R}^n$
- Topology: Proof that a finitely generated cone is closed
- Partitioning $\mathbb{R}^2$ into disjoint path-connected dense subsets
- Connectedness of the boundary
- When is a quotient by closed equivalence relation Hausdorff
- Conditions that ensure that the boundary of an open set has measure zero
- Every path has a simple “subpath”

Yes. Recall that any two countable dense linearly ordered sets without endpoints are order isomorphic, and hence homeomorphic. (See this question for a discussion, or this Wikipedia article for a proof.)

Let $S$ be the subset of $\mathbb{Q}$ consisting of the fractions with odd denominator (in lowest terms), and let $T$ be the complement of $S$. Then $S$ and $T$ are both countable, dense, and have no endpoints.

Let $U = \mathbb{Q}\cap (-\infty,\sqrt{2})$, and let $V = \mathbb{Q}\cap(\sqrt{2},\infty)$. Again, $U$ and $V$ are countable and dense, and have no endpoints.

Let $f\colon \mathbb{Q}\to\mathbb{Q}$ be a function that maps $U$ homeomorphically to $S$, and maps $V$ homeomorphically to $T$. Then $f$ is continuous and bijective, but its inverse is certainly not continuous, so $f$ cannot be a homeomorphism.

- If $A \in \mathbb{C}^{m\times n}$ is full-column rank matrix, then is rank($AB$) = rank ($BA$) = rank($B$)?
- Number of Ways to Fill a Matrix with symbols subject to Weird contsraint.
- Why do we not have to prove definitions?
- Definability in a given structure
- Evaluating $\sqrt{6+\sqrt{6+\cdots}}$
- Verification for the solution following differential equation!
- Conditions under which $a+b+c$ divides $1-abc$
- $x,y$ are integers satisfying $2x^2-1=y^{15}$, show that $5 \mid x$
- at least one element fixed by all the group
- Prove an algorithm for logarithmic mean $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{a_0-b_0}{\ln a_0-\ln b_0}$
- Proof of $\limsup\sin nx=1, n\rightarrow \infty \forall x\in \mathbb{R}.$
- General Information about Eigenvalues for an 3×3 symmetric matrix
- Computing the Hilbert class field
- Show that $x(x+1) = y^4+y^3+ay^2+by+c$ has a finite number of positive integral solutions.
- Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$