Epsilon-neighborhood

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In mathematics, the ε-neighborhood (or epsilon-neighborhood) of a set A \subseteq M, where (M,d) is a metric space, is the set of all points of M whose distance to some point of A is less than \varepsilon > 0. It is customarily denoted by A^\varepsilon. One may write

A^{\varepsilon} := \{ p \in M | \exists q \in A \mathrm{\,s.t.\,} d(p, q) < \varepsilon \} = \bigcup_{p \in A} B_{\varepsilon} (p)

where B_{\varepsilon} (p) is the open ball of radius \varepsilon centered at p.

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