Definition:Neighborhood (Metric Space)

Definition

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:

$N_\epsilon \left({a}\right) := \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\}$

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

Neighborhood in Pseudometric Space

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

The $\epsilon$-neighborhood of $a$ in $M$ is defined in exactly the same way as for a metric space:

$N_\epsilon \left({a}\right) := \left\{{x \in A: d \left({x, a}\right) < \epsilon}\right\}$

Radius

In $N_\epsilon \left({a}\right)$, the value $\epsilon$ is referred to as the radius of the ball.

Notation and Naming

There are various names and notations that can be found in the literature for this concept, for example:

• Open $\epsilon$-ball neighborhood of $a$ (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 of $a$;
• Open $\epsilon$-ball centered at $a$;
• $\epsilon$-ball at $a$.

Rather than say epsilon-ball, as would be technically correct, the savvy modern mathematician will voice this as the conveniently bisyllabic e-ball, to the apoplexy of his professor. And I don't believe anybody actually says open epsilon-ball neighborhood very often, whatever opportunities to do so may arise. Life is just too short.

Linguistic Note

The UK English spelling of this is neighbourhood.

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{III}$
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Definition $2.3.1$