Definition
Epsilon Neighbourhood (Vector)
The
-neighbourhood of a vector is the set of all vectors whose distance from is strictly less than : where
denotes the Euclidean norm. Geometrically, this set corresponds to an open ball centred at
with radius . For , it is an open interval; for , it is an open disk; and for , it is an open solid ball.
Significance
The epsilon neighbourhood is a fundamental tool for defining limits and continuity in multivariable calculus. It generalises the concept of proximity from the real line to higher-dimensional spaces. A set