Apparently bullet (2) hypothesis of definition of Sheaf only requires basic open families of sets. Proof: https://math.stackexchange.com/questions/4990717/can-we-apply-the-hypothesis-of-bullet-2-in-the-definition-of-sheaf-using-mere
@Shaun topology is very interesting, fun, and when you combine it with other fields, you get new theorems rather quickly.
PS. Found major issues with my previous approach (that I emailed you). Now I think I've got it (yet again). It uses a freaking sheaf this approach, who would have thought! I'm new to sheaf theory. But basically you can glue together some stuff with them. I also came up with a new topology which is courser than Furstenberg's, and hasn't been written about AFAIK.
*coarser
That simply means the open sets of the new one are strict subset of the open sets of the larger one (Furstenberg's full topology).
You must study some topology at some point! Topological groups, etc. are right in your area.
@AlessandroCodenotti If $X$ is totally disconnected compact Hausdorff, $|X|\geq \mathfrak{c}$, does there exists a a connected Hausdorff $Y$, $|Y|\geq 2$, and a continuous surjection $f:X\to Y$?
Let $X$ be the one-point compactification of discrete space of size $\mathfrak{c}$ and suppose there exists a surjection $f:X\to Y$ where $Y$ is connected Hausdorff and non-trivial, then there exists surjection $g:Y\to [0, 1]$, so we can assume $Y = [0, 1]$. By taking $\infty\in X$ and considering neighbourhood of $U$ of $f(\infty)$ not including infinitely many points, $f^{-1}(U)$ must be cofinite yet its complement must somehow surject onto an infinite set
And to answer another question of mine, suppose $X$ is the one-point compactification of discrete space, $\alpha > 0$ a non-limit ordinal, then every compactification of $X\times \omega_\alpha$ is zero-dimensional. Indeed, if $Z$ is such compactification that isn't, then there is closed connected non-trivial $Y\subseteq Z\setminus (X\times \omega_\alpha)$ and since $Y$ surjects onto $[0, 1]$, so does $Z\setminus (X\times \omega_\alpha)$ and hence $X\times \{\omega_\alpha\}$, which is impossible
forgot condition: also $|X| < \aleph_\alpha$
This shows that the property "every compactification of $X$ is zero-dimensional" doesn't restrict the cardinality of $\beta X \setminus X$ in any way, it can be as large as we want
I'm losing hope that any nice characterization of such spaces exists
Well, for locally compact spaces at least, this is equivalent to "$\beta X\setminus X$ doesn't surject onto $[0, 1]$" which is something
@SineoftheTime Out of curiosity I tried to solve this exercise: $\int_{+{\partial{D}}} \tan(z) \text{dz}$ with $|z| \leq 2$, then I rewrote the integral as $\int_{+{\partial{D}}} \frac{\sin(z)}{\cos(z)} \text{dz}$ singularity $\cos(z) = 0$ when $z = π/2 + nπ$ , then $|π/2 + nπ| ≤ 2$
I found $n = -1$ and $n = 0$, so $z = ± π/2$
The limit Is $\lim_{z \to\ +\frac{π}{2}} \frac{\sin(z) (z - π/2)}{\cos(z)}$
@BenSteffan can I link you to my most recent question. It's rather simple. I don't use much properties of sheaves, but need to satisfy some hypotheses using my local sections. Once I can prove that, then the global section must be continuous and unique.
You only see the best of them (in literature), so we get a skewed perspective
If I can get this proof to work, there's a way to show that if at least two $2k$-separated primes exist, then there must be infinitely many such pairs. But my paper would focus just on twin primes, leaving the other cases open.
$0$-separated primes are the primes themselves!
Zhang won some monetary awards for his work. I at least want to be an honorary professor and travel the world to teach / consult. If the journals shaft me, I'll just self-publish, even write a book on elementary topological NT.
It's great. It's not an official prize problem like the Clay Institute $1M problems, but it's still considered an unofficial one.
I'm no crackpot. However, I'm not in school, so it feels like there's a huge wall in front of me
I would like an MSE alternative website, where prize problem questions don't get deleted. Proofs can be peer-reviewed / checked, and credit automatically attributed to the right person. I gave up on my huge commutative diagram site project. This site idea would be simpler - no logic backend, just user edited content
It's ridiculous how oppressive the site is against anything that isn't a homework problem, yet do my homework is also not allowed. It's a contradiction obviously
It's hampering future mathematicians from doing what they do best. And we're doing it for free. Still they try to control us
Perhaps it's the best policy for high-quality content, but just wish there were something more suitable for mathematicians to actually "do math" / participate / be creative. Rather than just for learning what's already been done in math.
@Ben turns out the maps in the saturated class generated by the horn inclusions $\Lambda_k^n\rightarrow\Delta^n$ for $n\ge2$ and $0\le k\le n$ are precisely the anodyne maps that are bijective on vertices satisfying the additional condition that the edges of the domain generate all edges in the target under composition and cancellation, which is surprisingly intuitive once you think about it
