Intereting Posts

Centralizer/Normalizer of abelian subgroup of a finite simple group
Hilbert's Nullstellensatz: intersection over maximal ideals?
Intermediate fields of a finite field extension that is not separable
Complex integral help involving $\sin^{2n}(x)$
class function question
Who discovered this number-guessing paradox?
Integrable but not differentiable function
Why does the non-negative matrix factorization problem non-convex?
Probability of combinations of beads on cut necklaces (mass spectrometry physics problem)
Nets and Convergence: Why directed indices?
A normal subgroup intersects the center of the $p$-group nontrivially
Please suggest a functional analysis book to refresh my knowledge
Is there an elementary way to see that there is only one complex manifold structure on $R^2$?
Number of ways of walking up $6$ steps taking $1, 2,$ or $3$ at a time, and a recurrence relation
Find the value of $x_1^6 +x_2^6$ of this quadratic equation without solving it

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.

- $G_\delta$ sets
- Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition
- where can i find this A.H. Stone's theorem proof?
- Making sense out of “field”, “algebra”, “ring” and “semi-ring” in names of set systems
- Examples of metric spaces which are not normed linear spaces?
- Relations and differences between outer/inner limit and Kuratowski limsup/liminf

- How to Axiomize the Notion of “Continuous Space”?
- In an N-dimensional space filled with points, systematically find the closest point to a specified point
- partial converse of existence of covering spaces
- Can two topological spaces surject onto each other but not be homeomorphic?
- A question about connected sets in $\mathbb{R}^2$
- What can we say about a locally compact Hausdorff space whose every open subset is sigma compact?
- When do weak and original topology coincide?
- Vector Bundles Over a Manifold
- Are the only sets in $\mathbb{R^1}$ which are both open and closed $\mathbb{R^1}$ and $\emptyset$?
- Does Seperable + First Countable + Sigma-Locally Finite Basis Imply Second Countable?

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.

- Characteristic of a field is $0$ or prime
- Let $X$ and $Y$ be independent, exponentially distributed with mean 1. Show $\dfrac{X}{X+Y}$ is uniformly distributed in the interval $(0, 1)$.
- a neighbourhood of identity $U$ generates $G$ where $G$ is a connected lie group
- Polynomials calaculatung
- Convergence of random variables in probability but not almost surely.
- What are some strong algebraic number theory PhD programs?
- Reasons behind the field axioms : $1 \ne 0$ and $1/0$ not defined
- Riemann integrable function
- Proof of $f = g \in L^1_{loc}$ if $f$ and $g$ act equally on $C_c^\infty$
- Distinctness is maintained after adding some element to all sets
- Guessing a subset of $\{1,…,N\}$
- Irreducible representation which is not induced one
- Probability of picking all elements in a set
- How do I solve this equation involving a logarithm?
- If $f$ is an entire function with $|f(z)|\le 100\log|z|$ and $f(i)=2i$, what is $f(1)$?