Definition:Open Ball

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

Let $a \in A$.

Let $\epsilon \in \R_{>0}$ be a strictly positive real number.

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


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

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

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 neighborhood the notation $N_\epsilon \left({a}\right)$, $N \left({a, \epsilon}\right)$ or $N \left({a; \epsilon}\right)$ are often seen)
 * Spherical neighborhood of $a$
 * Open sphere at $a$
 * Open $\epsilon$-ball centered at $a$
 * $\epsilon$-ball at $a$.

The notation $B \left({a; \epsilon}\right)$ can be found for $B_\epsilon \left({a}\right)$, particularly when $\epsilon$ is a more complicated expression than a constant.

Similarly, some sources allow $B_d \left({a; \epsilon}\right)$ to be used for $B_\epsilon \left({a; d}\right)$.

It needs to be noticed that the two styles of notation allow a potential source of confusion, so it is important to be certain which one is meant.

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 at least one person on this site doesn't believe anybody actually says open epsilon-ball neighborhood very often, whatever opportunities to do so may arise. Life is just too short.

The term neighborhood is usually used nowadays for a concept more general than an open ball: see Neighborhood (Metric Space).

Also see

 * Open Ball is Open Set, in which it is shown that $B_{\epsilon} \left({a}\right)$ is open in $M$.
 * Neighborhood
 * Deleted Neighborhood