Intereting Posts

Proving that $ 30 \mid ab(a^2+b^2)(a^2-b^2)$
Ring isomorphism (polynomials in one variable)
Why is it that $\mathscr{F} \ne 2^{\Omega}$?
Fibonacci Recurrence Relations
Projection of v onto orthogonal subspaces are the those with minmum distance to v?
What is the equation family of the projectile-motion-with-air-resistance eqn?
Finding the matrix for a linear transformation on a vector space when the basis changes
Determining the cardinality of $SL_2 (F_3) $
Derivative of $ 4e^{xy ^ {y}} $
Multichoosing (Stars and bars)
Hölder continuous but not differentiable function
A uniform bound on $u_n$ in $L^\infty(0,T;L^\infty(\Omega))$
Bounds on the average of the divisors of natural numbers.
Completeness of ${C^2}$ with under a specific metric
What is a closed form for ${\large\int}_0^1\frac{\ln^3(1+x)\,\ln^2x}xdx$?

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle?

Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of $\mathbb{R}$.

What about if we change the topology by consideration of $p-$adic topology on rational numbers?

- Multidimensional Hensel lifting
- What are the quadratic extensions of $\mathbb{Q}_2$?
- How do you take the multiplicative inverse of a p-adic number?
- Profinite and p-adic interpolation of Fibonacci numbers
- Tensor product of a number field $K$ and the $p$-adic integers
- The $p$-adic integers as a profinite group

- Resolution of Singularities: Base Point
- Local solutions of a Diophantine equation
- What is the group structure of 3-adic group of the cubes of units?
- Proving $\sqrt{2}\in\mathbb{Q_7}$?
- Show that $y^2=x^3+1$ has infinitely many solutions over $\mathbb Z_p$.
- integral cohomology ring of real projective space
- Does the exceptional Lie algebra $\mathfrak{g}_2$ arise from the isometry group of any projective space?
- $p^a\mid f(v) \implies p^a\mid f(w)$ in $\mathbb Z$
- Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$
- An automorphism of the field of $p$-adic numbers

A space is called *ultraparacompact* if every open cover can be refined to a cover by disjoint open sets. Note that clearly any fiber bundle on an ultraparacompact space is trivial (just apply the definition to an open cover on which it is trivial). So it suffices to show that $\mathbb{Q}P^n$ is ultraparacompact (with either the standard topology or the $p$-adic topology). More generally, I will prove the following:

Theorem: Any countable space with a basis of clopen sets is ultraparacompact.

Proof: Let $Q=\{q_n\}$ be a countable space with a basis of clopen sets and let $\mathcal{U}$ be an open cover. Define a refinement of $\mathcal{U}$ by induction. First, let $V_1$ be any clopen set containing $q_1$ which is contained in some element of $\mathcal{U}$. Next, if $q_2\in V_1$, let $V_2=V_1$; otherwise, let $V_2$ be a clopen set containing $q_2$ which is contained in some element of $\mathcal{U}$ and disjoint from $V_1$. Continue by induction, adding clopen sets contained in some element of $\mathcal{U}$, containing the next $q_n$, and disjoint from the clopen sets we’ve chosen so far. In the end, we get an open cover $\{V_n\}$ refining $\mathcal{U}$ consisting of disjoint clopen sets.

- $f,\overline f$ are both analytic in a domain $\Omega$ then $f$ is constant?
- If $\int(f_n) \rightarrow \int(f)$ then $\int(|f_n-f|) \rightarrow 0$ for $f_n \rightarrow f$ pointwise
- How many cycles, $C_{4}$, does the graph $Q_{n}$ contain?
- $a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$…
- Derivative of ${x^{x^2}}$
- Character on conjugacy classes
- Result of the product $0.9 \times 0.99 \times 0.999 \times …$
- Let $f:A \to B$ and $g:B\to A$ be arbitrary functions.
- Sum of all elements in a matrix
- Cellular Boundary Formula
- book with lot of examples on abstract algebra and topology
- $I:=\{f(x)\in R\mid f(1)=0\}$ is a maximal ideal?
- Baby Rudin Theorem 1.20 (b) Proof
- A Tough Problem about Residue
- A curious identity on sums of secants