neighborhood
(abbrev. nbd), n. 1. also called ɛ-neighborhood. (in a Euclidean or metric space) the open set of all points with distance from a given point strictly less than a specified value; the set of points
written N(ɛ, a). An open ɛ-neighborhood is also called an open ball or open sphere. In this notation, a function is said to have a limit as x tends to a if there is a p such that
for every ɛ > 0, there exists a δ > 0 such that
Compare epsilon-delta notation.
2a. more generally, any set in a topological space that contains an open set that contains the given point; in a Euclidean or metric space, this is any set that contains an ɛ-neighborhood, so that every ball is an open neighborhood, but an open neighborhood is not necessarily a ball. Some authors avoid this usage and restrict the unqualified term to open neighborhoods. A closed ɛ-neighborhood of a point a in a metric space is defined by
{x : d(x, a) ≤ ɛ}.
b. punctured neighborhood. derivatively, a neighborhood of a point from which the given point itself has been deleted; that is, a punctured ɛ-neighborhood of a, written N'(ɛ, a), is N(ɛ, a)\{a}. 3. (of infinity) a neighborhood of an ideal point added in a compactification. Thus ]-r, ∞] is an open neighborhood of +∞ on the real line.
{x : d(x, a) < ɛ},
written N(ɛ, a). An open ɛ-neighborhood is also called an open ball or open sphere. In this notation, a function is said to have a limit as x tends to a if there is a p such that
for every ɛ > 0, there exists a δ > 0 such that
f(x) ∈ N(ɛ, p) for all x such that x ∈ N(δ, a).
Compare epsilon-delta notation.
2a. more generally, any set in a topological space that contains an open set that contains the given point; in a Euclidean or metric space, this is any set that contains an ɛ-neighborhood, so that every ball is an open neighborhood, but an open neighborhood is not necessarily a ball. Some authors avoid this usage and restrict the unqualified term to open neighborhoods. A closed ɛ-neighborhood of a point a in a metric space is defined by
{x : d(x, a) ≤ ɛ}.
b. punctured neighborhood. derivatively, a neighborhood of a point from which the given point itself has been deleted; that is, a punctured ɛ-neighborhood of a, written N'(ɛ, a), is N(ɛ, a)\{a}. 3. (of infinity) a neighborhood of an ideal point added in a compactification. Thus ]-r, ∞] is an open neighborhood of +∞ on the real line.