What's a "deleted neighborhood"? (other than very very confusing)

The first contains $x$ (as the distance is allowed to be $0$) while the second excludes $x$.

Also, the absolute value is NOT always positive. It is always non-negative. $0$ matters.


It is also called punctured. It's the neighborhood of a point minus the point itself. It's very useful as a tool to know whether a point is a limit point of a set. A point is a limit point of a set when any deleted neighborhood of such a point has a non-empty intersection with the set.

Tags:

General Topology

Real Analysis

Real Numbers

Related

Between which two integers does $\sqrt{2017}$ fall? Does $(\cos t , \sin (\sqrt{2}t))$ traces all the points of a square. Prove that $\lim\limits_{x \to 1} \frac{x+2}{x^2+1} = \frac{3}{2}$ How to prove $\phi=1+\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{F_nF_{n+1}}$ Expected number of items looked at $a^2+b=b^{1999}$ how many integer pairs $(a,b)$ are there that satisfy this equation. For some integer $q$, $q^2 - 5$ is divisible by all of the following EXCEPT which of the following? If $a,b$ are positive integers and $x^2+y^2\leq 1$ then find the maximum of $ax+by$ without differentiation. $66$ points in $100$ shots. How to determine the $\lim_{n\to \infty} \frac{1+2^2+\ldots+n^n}{n^n}=1$. Density of $X-Y$ where $X,Y$ are independent random variables with common PDF $f(x) = e^{-x}$? Apparent inconsistency of Lebesgue measure

Recent Posts