So in the Alexandroff case you can construct a minimal basis by looking at each point $x$ and taking the intersection of open sets containing it, calling that $U_x$
(I have a question on main with a nice answer asking for reasonable properties ensuring that a space has a minimal basis if you want to look at it after thinking about this problem for a while)
For sure. I have to understand the "section of the etale space" business better to actually understand why you're claiming the zero set is open though.
Okay that's only a preorder, actually, $T_0$ makes it into an order since $U_x = U_y$ iff $x=y$. So now you have a partial order. Reverse, gimme a partial order
Then define $U_x = \{y:y\le x\}$
And the generated topology is Alexandroff and $T_0$
> Think of the sheaf of continuous functions on a top. space. A section is a continuous function, so its zero set is closed. On the other hand, a section is a local homeomorphism, so its zero set is open.
@TedShifrin sur le plateau de Saclay, là où est situé Polytechnique depuis ~1974. Le gouvernement veut rassembler les grandes écoles et les universités à cet endroit afin de former une sorte de campus unifié (et remonter dans les classements internationaux)
> Think of the sheaf of continuous functions on a top. space. A section is a continuous function, so its zero set is closed. On the other hand, a section is a local homeomorphism, so its zero set is open.
@LeakyNun As an example, if I take the sum of all reciprocal 100th powers of integers, I know that this sum will be $\pi^{100}$ times a rational number.
@TedShifrin Are you arguing something like, if a section $s$ is so that $s(p) = 0$ at $p \in X$, then the germ of $s$ at $p$ is zero, so $s$ has to be zero on an open set on $X$ around $p$?
@TedShifrin I guess the point is that the zero set in the topological context need not be the same as the sheaf context. The global function $f : X \to \Bbb R$ corresponding to $s$ can be zero at $p$ without it's germ being zero at $p$. Is that it?
There are many amazing results about amorphous sets. However, I have yet to find one actual construction. Can an amorphous subset of $\Bbb R$ be explicitly constructed, assuming the negation of choice?
@LeakyNun I'm pretty sure by any reasonable metric, proof theory is a part of math itself, just that the areas to which it applies are less other areas of math that aren't obviously related and more CS theory
Hey guys/men and women... may I ask you all a personal question? (Sorry for my interuption). Why did you choose to study math (I suppose most of you did or are studying it).
Haha, I do not. I haven't finished highschool yet, or "gymnasium" as it is called in Sweden. But I think I will study math after finishing highschool. I find it beautiful, even the most intricate of details in it can be very beautiful. This may sound VERY strange to some ears, but math could be compared to love sometimes. It is interesting from time to time, and it is peaceful to work with math. It can also be very challenging, which I like @LeakyNun
The problem here seems to be that he don't know what argmin means. He want A to be the same size of formula and I got -ve vote. Even by replying to his comment asking what did he meant. I actually sent that the formula is from a published paper and send the link. I flagged the question for a moderator to have an intervention then I got another -ve vote in 1 minute. So I deleted the question.
I do believe that there exist people who mean it when they say that "math is beatiful". It is not something sterile, not only a tool to be picked up at certain times.
@Mour_Ka, this place can be a frustrating hell at times.
It is fine, I just want the question to be correct if it is not. I also want at least a guidance that I can start from. Seems I will ask people in office after all. But Is there a way that I can involve a moderator to help me remove the question totally and its negative contributions (votes and comments)
Suppose $M$ and $N$ are smooth manifolds with boundary and $F : M \to N$ is a diffeomorphism. Then $f(\partial M) = \partial N$, and $F$ restricts to a diffeomorphism from $\text{Int} M$ to $\text{Int} N$
This is exercise $2.19$ in *Introduction to Smooth Manifolds by John Lee*. The hint is to use the theorem on the smooth invariance of the boundary of a manifold to prove this
Theorem : Suppose $M^n$ is a smooth manifold with boundary and $p \in M$. If there exists a smooth chart $(U, \psi)$ for $M$ such that $\psi[U] \subseteq H^n$ and $\psi(p) \in \partial H^n$, then for every other smooth chart $(U, \phi)$, with $\phi[U] \subseteq H^n$, we have $\phi(p) \in \partial H^n$
@Perturbative Consider $\operatorname{Aut}(M)$ and $\operatorname{Hom}(N)$. Clearly, they are isomorphic, now, use sheaf theory to show that $\operatorname{Int}M$ and $\delta N$ have the same section. After that, realize that I have absolutely no idea what I'm talking about.
@BalarkaSen Well a boundary chart $(U, \psi)$ around $p \in \partial M$, is gonna map a neighbourhood $U$ of $p$ to $\psi[U]\subseteq H^m$ with $\psi[U] \cap \partial H^m \neq \emptyset$
$$-\log (1.8* 10^{-5})= 5 - \log 1.8$$ Please verify if you can.. I don't know logs properly, will study them in Maths later. Right now I am dealing with $pH$ calculation problems so I need basic log knowledge.
@Perturbative Right. So locally the boundary looks like $\Bbb R^{n-1} = \partial \Bbb H^n \subset \Bbb H^n$. If $f$ maps $p \in \partial M$ to $f(p)$ diffeomorphically, what must a chart around $f(p)$ look like?
@BalarkaSen In essence what I tried to do was complete a commutative diagram for $F$, and handle the cases for the dimensions of $M$ and $N$ respectively
@BalarkaSen There would be two possibilities for a chart around $f(p)$, $f(p)$ would be the domain of an interior chart or the domain of a boundary chart, either mapping to $\mathbb{R}^n$ or $H^n$ with nonempty intersection of $\partial H^n$ respectively
@Perturbative Right. But since $p$ is a boundary point in $M$, you can take a boundary chart $(U, \varphi)$ around $p$ which maps $\varphi : U \to \Bbb H^n$ diffeomorphically to an open set on the closed upper half plane hitting the boundary.
With the diffeomorphism $f : M \to N$, you should be able to produce a boundary chart for $f(p)$.
