Epsilon-neighborhood

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In mathematics, the ε-neighborhood (or epsilon-neighborhood) of a set A ⊆ 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 ε > 0. It is customarily denoted by Aε. 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 Bε(p) is the open ball of radius ε centered at a point p of M.

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