Gleason's theorem: Difference between revisions
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In [[mathematics]], the '''''ε''-neighborhood''' (or '''epsilon-neighborhood''') of a [[set (mathematics)|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.<ref name=Barile>{{cite web|last=Barile, Margherita and Weisstein, Eric W.|title=Neighborhood|url=http://mathworld.wolfram.com/Neighborhood.html|publisher=MathWorld|accessdate=21 September 2013}}</ref> It is customarily denoted by ''A''<sup>''ε''</sup>. One may write | |||
:<math>A^{\varepsilon} := \{ p \in M | \exists q \in A \mathrm{\,s.t.\,} d(p, q) < \varepsilon \} = \bigcup_{p \in A} B_{\varepsilon} (p)</math> | |||
where ''B''<sub>''ε''</sub>(''p'') is the [[open ball]] of radius ''ε'' centered at a point ''p'' of ''M''. | |||
==References== | |||
{{reflist}} | |||
{{DEFAULTSORT:Epsilon-Neighborhood}} | |||
[[Category:General topology]] | |||
[[Category:Metric geometry]] | |||
Revision as of 07:27, 4 December 2013
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.[1] It is customarily denoted by Aε. One may write
where Bε(p) is the open ball of radius ε centered at a point p of M.
References
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