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:


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

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 set in the context of metric spaces, 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 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.