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
Neighborhood, set-builder
notation, interval notation
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