Definition:Neighborhood (Metric Space)

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This page is about neighborhoods in the context of metric spaces. For other uses, see Definition:Neighborhood.

Definition

Let $M = \left({A, d}\right)$ be a metric space.

Let $S \subseteq A$ be a subset of $A$.


Let $x \in S$.

Let there exist $\epsilon \in \R_{>0}$ such that the open $\epsilon$-ball at $x$ lies completely in $S$, that is:

$B_\epsilon \left({x}\right) \subseteq S$

Then $S$ is a neighborhood of $x$ in $M$.


Also see

  • Results about neighborhoods can be found here.


Linguistic Note

The UK English spelling of neighborhood is neighbourhood.


Sources