Neighborhood of a point (real number)

How can we prove that every real number has infinitely many neighbourhoods?
I know that it is true because we can consider a symmetric epsilon neighborhood of a point and there are infinitely many number of such epsilons.
But how can we prove it rigorously?

ag.algebraic geometry – Holomorphic tubular neighborhood of divisors at infinity

For the discussion of holomorphic tubular neighborhoods and some criteria for their existence see this question.

Let $X$ be a smooth quasi-projective variety over $mathbb{C}$. Hironaka tells us that there exists a smooth projective compactification $X subset bar{X}$, such that $D =bar{X}backslash X$ is a strictly normal crossing divisor. Let $D=cup_{i}^N D_i$ be its decomposition into irreducible smooth divisors. What is the obstruction to finding such $bar{X}$, such that each $D_i$ has a holomorphic tubular neighborhood in $bar{X}$?

Example 1: Let $S$ be a smooth projective variety, $E$ a vector bundle on $S$. The space $X=text{Tot}(E)$ is compactified to $bar{X}=mathbb{P}_S(Eoplus mathcal{O}_S)$ with the divisor at infinity $D =mathbb{P}_S(E)subsetbar{X}$. Its normal bundle should be $mathcal{O}_{mathbb{P}(E)}(1)$ and one should be able to find a holomorphic tubular neighborhood by constructing one in each fiber and gluing together.

Example 2: Using the answer by Joey, I have convinced myself that if the divisor $D_i$ is $mathbb{P}^n$ for some $ngeq 2$ and $n+1=text{dim}(X)$, then it also has such a neighborhood. If $n=1$, then one needs additionally that $(D_i)^2<0$.

The above examples make me believe that it could be always achievable, as each blow up at a smooth point introduces a new $mathbb{P}^{n}$ with normal bundle $mathcal{O}(-1)$.

real analysis – The existence of inverse function of “rotated” $f(x)=x^{3} sin{frac{1}{x}}$ on the neighborhood of x=0?

I’m a bit confused about the existence of inverse function of a function. Let’s consider the “rotated” version of $f(x)=x^{3} sin{frac{1}{x}}$, say, rotate it counterclockwise around the origin through a small enough angle $theta$, to yield a wave that straddles a line $y=kx$, where $k=tan{theta}$.

Then we see since the rotation angle is very small, any straight line $y=c$ must intercept the figure of the function at least 2 points(when c is small enough), thus the inverse function on any neighborhood of x=0 cannot exist.

But by inverse function theorem, $f$ is continuously differentiable, $f^{prime}=k>0$, and $f(0)=0$, it should have inverse function on a neighborhood of x=0.

Why?

html: what's a good way to embed the FedWiki neighborhood concept on my own page?

When you open a new link on the Federated Wiki, the old pages don't disappear. Instead, the new page opens next to the previous one. As you open more links, the older pages move further to the right. In this way, you can always view at least two of the pages you previously visited to provide context for the current page. This is called a neighborhood.

I would love to include a similar concept on my own page. What is a good way to do this?

(What I have in mind is a bunch of static html sites that are connected via links and displayed in this type of neighborhood structure.)

enter the image description here

array: get subpictures of a binary image with specified neighborhood

Let's say I have this binary image (an array with 1 and 0):

start = {{0, 1, 1, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}, {1, 1, 0, 0}};
i = ArrayPlot(CellularAutomaton(<|"OuterTotalisticCode"->224,"Dimension"->2,"Neighborhood"->9|>,{start,0},{{{200}}}),Frame->False,ColorRules->{1->White,0->Black})

enter the image description here

Then I can extract the subpictures, eg. by using:

subimages =  ComponentMeasurements(i, "Image", All, "PropertyAssociation")("Image")

enter the image description here

This gives me 13 subpictures that have a black pixel in the middle. (Please note that the above blank entries are white pixels and therefore the result is correct.)

But now I'd like to count subpictures with two black pixels in the middle as a sub picture. More precisely, I would like to get just one result with 9 sub-images grouped like this:

enter the image description here

Is there an obvious way to do it in Mathematica?

openvpn: how much bandwidth and RAM x users does a VPS need to be used as a neighborhood VPN?

I have to rent a VPS to host a neighborhood VPN for around 30-50 users.

The VPN will be a complete tunnel and in addition to being used for browsing, social networks, shopping, etc., it will also be used to download files or stream videos.

How much RAM and bandwidth will the VPS have to guarantee a decent connection for each user?

New kid in the neighborhood

Hi guys, I'm new here …

at.algebraic topology – When is a deformation retraction of its open neighborhood multiple limits?

For this question, a multiple is a locally superior Euclidean Hausdorff space; Paracompaction or second accounting is not assumed, and there may be limits.

Leave $ M $ being a distributor What are the sufficient conditions for it to exist open? $ U subset M $ such that $ partial M subset U $ Y $ U $ (Strongly?) Deformation retracts over $ partial M $? (Could this be true for all multiples?)

Clearly $ partial M $ be trapped in $ M $ it is a fairly general enough condition (this includes multiple paracompacts). However, I would be interested to know if there is a sufficient condition more general than this.

general topology: why is the neighborhood a topological space?

I am a total dilettante in topology, therefore, my apologies for this question. Wikipedia writes that:

"… every point of a northmultidimensional has a neighborhood that is homeomorphic to the space of Euclidean dimension north".

As I understand it, a homeomorphism is a kind of isomorphism between topological spaces. So, the neighborhood must be a topological space. What is this space (that is, its open sets or base)?

Issues – Solve the problem in the menu module outside the canvas for the Neighborhood subtopic

I need help to change a subtopic of Barrio (v 8.x-4.22) of Drupal 8
This is what I have done so far:

The menu appears at the bottom of the page.This is how it looks now
I don't know where the problem is and how to solve it.
I tried to enable the debugging of twigs (from here-> https://www.drupal.org/docs/8/theming/twig/debugging-twig-templates) and I think the problem is in the menu settings outside the canvas , but I'm not sure how to change them.