Definition:Open Ball/Also known as

Terminology for Open Ball
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 $\map {N_\epsilon} a$, $\map N {a, \epsilon}$ or $\map N {a; \epsilon}$ are often seen)
 * Spherical neighborhood of $a$
 * Open sphere at $a$
 * Open $\epsilon$-ball centered at $a$
 * $\epsilon$-ball at $a$.

Some sources use the  symbol $\varepsilon$ instead of the   which is $\epsilon$.

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

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

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.

Some sources use $\epsilon B$ as a convenient shorthand for $B_\epsilon$, allowing it to be understood that $B$ is an open unit ball, but this is idiosyncratic and non-standard.

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 contributor to this site does not believe that nobody 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).