Real Neighborhoods

Why

Often when speaking of a set, we are interested in speaking of those elements which are close to it.

Definition

Let $x \in \R ^d$ A subset $N \subset \R ^d$ is a neighborhood of $x$ if there is a $\delta > 0$ such that $B(x, \delta ) \subset N$. The set $\mathcal{N} _a$ of neighboords of $x$ is called the complete system of neighborhoods of the point $a$.

We interpret a neighborhood of a point $x \in X$ as a set containing all the points of $X$ that are sufficiently close to $a$. A neighborhood of $x$ “encloses” $x$ by virtue of it containing an open ball about $a$.¹

1. Future editions will continue to treatment, including pointing out that an open ball at $x$ is a neighborhood of $x$ and of all elements in the ball. ↩︎