Definition:Neighborhood (Analysis)

Topology
Let $$\left({X, \vartheta}\right)$$ be a topological space.

Neighborhood of a Set
Let $$A \subseteq X$$ be a subset of $$X$$.

A neighborhood of $$A$$, which can be denoted $$N_A$$, is any subset of $$X$$ containing an open set which itself contains $$A$$.

That is:
 * $$\exists U \in \vartheta: A \subseteq U \subseteq N_A \subseteq X$$

Neighborhood of a Point
The set $$A$$ can be a singleton, in which case the definition is of the neighborhood of a point.

Let $$z \in X$$ be a point in a $$X$$.

A neighborhood of $$z$$, which can be denoted $$N_z$$, is any subset of $$X$$ containing an open set which itself contains $$z$$.

That is:
 * $$\exists U \in \vartheta: z \in U \subseteq N_z \subseteq X$$

Open Neighborhood
If $$N_A \in \vartheta$$, i.e. if $$N_A$$ is itself open in $$X$$, then $$N_A$$ is called an open neighborhood.

Some authorities require all neighborhoods to be open.

Elementary Properties

 * From this definition, it follows directly that $$X$$ itself is always a neighborhood of any $$A \subseteq X$$.


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

A set which is the neighborhood of all its points is open.

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.