Definition:Open Ball

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.

Also known as
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.

Also see

 * Deleted Neighborhood