Definition:Neighborhood (Analysis)
This page is about neighborhoods in the context of analysis. For other uses, see Definition:Neighborhood.
Contents
Definition
Real Analysis
Let $\alpha \in \R$ be a real number.
Open Subset Neighborhood
Let $N_\alpha$ be a subset of $\R$ which contains (as a subset) an open real set which itself contains (as an element) $\alpha$.
Then $N_\alpha$ is a neighborhood of $\alpha$.
Epsilon-Neighborhood
On the real number line with the usual metric, the $\epsilon$-neighborhood of $\alpha$ is defined as the open interval:
- $\map {N_\epsilon} \alpha := \openint {\alpha - \epsilon} {\alpha + \epsilon}$
where $\epsilon \in \R_{>0}$ is a (strictly) positive real number.
Topology
Let $T = \left({S, \tau}\right)$ be a topological space.
Neighborhood of a Set
Let $A \subseteq S$ be a subset of $S$.
A neighborhood of $A$, which can be denoted $N_A$, is any subset of $S$ containing an open set of $T$ which itself contains $A$.
That is:
- $\exists U \in \tau: A \subseteq U \subseteq N_A \subseteq S$
Neighborhood of a Point
The set $A$ can be a singleton, in which case the definition is of the neighborhood of a point.
Let $z \in S$ be a point in a $S$.
Let $N_z$ be a subset of $S$ which contains (as a subset) an open set of $T$ which itself contains (as an element) $z$.
Then $N_z$ is a neighborhood of $z$.
That is:
- $\exists U \in \tau: z \in U \subseteq N_z \subseteq S$
Metric Space
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 $M$.
Complex Analysis
A specific application of this concept is found in the field of complex analysis:
Let $z_0 \in \C$ be a complex number.
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.
The $\epsilon$-neighborhood of $z_0$ is defined as:
- $\map {N_\epsilon} {z_0} := \set {z \in \C: \cmod {z - z_0} < \epsilon}$
Also see
- Results about neighborhoods can be found here.
Internationalization
Neighborhood is translated:
In Dutch: | omgeving | |||
In French: | voisinage |
Linguistic Note
The UK English spelling of neighborhood is neighbourhood.