Epsilon-neighborhood

Epsilon-neighborhood

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 "A""ε". One may write

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

where "B""&epsilon;"("p") is the open ball of radius "&epsilon;" centered at a point "p" of "M".


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