**Deleted Neighborhood**

The proper name for a set such as {*x*: 0 < |*x* – *a*| < δ}. Deleted neighborhoods are encountered in the study of limits. It is the set of all numbers less than δ units away from *a*, omitting the number *a* itself.

Using interval notation the set {*x*: 0 < |*x* – *a*| < δ} would be (*a* – δ, *a*) ∪ (*a, a* + δ). In general, a deleted neighborhood of *a* is any set (*c*, *a*) ∪ (*a*, *d*) where *c* < *a* < *d*.

For example, one deleted neighborhood of 2 is the set {*x*: 0 < |*x* – 2| < 0.1}, which is the same as (1.9, 2) ∪ (2, 2.1).

**See also**

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