Definition:Open Ball/Also known as

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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 $N_\epsilon \left({a}\right)$, $N \left({a, \epsilon}\right)$ 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 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).