Question

check whether the set [−3,7 [∩]− 7,3

]is a neighbourhood of 2 or not.

Answer 1

First, let's visualize the intersection $[-3;7[\cup[-7;3[$ of given sets

Conclusion,

$$\boxed{[-3;7[\cup[-7;3[=[-3;3[}$$

Definition. Let $a$ be a point in $\mathbb{R}$ and let $\varepsilon >0$. The open inerval $\left(a-\varepsilon;a+\varepsilon\right)$ centered at $a$ is called the $\varepsilon-$ NEIGHBOURHOOD of $a$ and is denoted $J_\varepsilon(a)$. Notice that this neighbourhood consists of all numbers $x$ whose distance from $a$ is less than $\varepsilon$; that is, such that $|x-a|<\varepsilon$.

(From John M. Erdman. A Problems Based Course in Advanced Calculus)

As can be seen from the definition, the neighborhood must be symmetric with respect to the point. And in our case, the given set is not symmetrical with respect to the point x = 2, since

$$\rho(2;3)=|3-2|=1<5=|-3-2|=\rho(-3;2)$$

Conclusion,

$$\boxed{[-3;3[ \,\,-\,\,\text{is not a neighborhood of a point}\,\,x=2}$$