Definition:Neighborhood (Analysis)

Topology
Let $$x \in X$$ be a point in a topological space with topology $$\vartheta$$.

Then a neighborhood of $$x$$ is a set $$V \subseteq X$$ such that $$\exists U \in \vartheta$$ and $$x \in U \subseteq V$$.

Note that in this definition, $$X$$ itself is always a neighborhood of any $$x \in X$$.

It also follows that any open set of $$X$$ containing $$x$$ is a neighborhood of $$x$$.

Some authorities require $$V \in \vartheta$$, i.e. for $$V$$ to be open in $$X$$, but the definition given here is more common.

Metric Space
Let $$M = \left({A, d}\right)$$ be a metric space.

Let $$a \in A$$.

Let $$\epsilon \in \R: \epsilon > 0$$ be a positive real number.

The $$\epsilon$$-neighborhood of $$a$$ in $$M$$ is defined as:

$$N_\epsilon \left({a}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\}$$.

If it is necessary to show the metric itself, then the notation $$N_\epsilon \left({a; d}\right)$$ can be used.

From the definition of open, it follows that an $$\epsilon$$-neighborhood in a metric space $$M$$ is open in $M$.

There are various names and notations that can be found in the literature for this concept, for example:
 * Open $$\epsilon$$-ball neighborhood (and in deference to the word "ball" the notation $$B_\epsilon \left({a}\right)$$, $$B \left({a, \epsilon}\right)$$ or $$B \left({a; \epsilon}\right)$$ are often seen);
 * Spherical neighborhood;
 * Open $$\epsilon$$-ball;
 * $$\epsilon$$-ball.

Complex Analysis
A specific application of this concept is found in the field of complex analysis.

Let $$z_0 \in \C$$ be a complex number.

Let $$\epsilon \in \R: \epsilon > 0$$ be a positive real number.

The $$\epsilon$$-neighborhood of $$z_0$$ is defined as:

$$N_\epsilon \left({z_0}\right) \ \stackrel {\mathbf {def}} {=\!=} \ \left\{{z \in \C: \left|{z - z_0}\right| < \epsilon}\right\}$$.

In this context, a neighborhood is often referred to as an open disk (UK spelling: open disc).

Real Numbers
On the real number line with the usual metric, $$N_\epsilon \left({a}\right)$$ is the open interval $$\left({a - \epsilon \, . \, . \, a + \epsilon}\right)$$.

Comment
The UK English spelling of this is "neighbourhood".