Neighbourhood (mathematics)

If $X$ is a topological space and $p$ is a point in $X,$ then a neighbourhood{{sfn|Willard|2004|loc=Definition 4.1}} of $p$ is a subset $V$ of $X$ that includes an open set $U$ containing $p$,

$$p\; \backslash in\; U\; \backslash subseteq\; V\; \backslash subseteq\; X.$$

This is equivalent to the point $p\; \backslash in\; X$ belonging to the topological interior of $V$ in $X.$

The neighbourhood $V$ need not be an open subset of $X.$ When $V$ is open (resp. closed, compact, etc.) in $X,$ it is called an {{visible anchor|open neighbourhood|open neighborhood}}{{cite book |last=Dixmier |first=Jacques |others=Translated by Sterling K. Berberian |year=1984 |title=General Topology |url=https://archive.org/details/generaltopology0000dixm |url-access=registration |series=Undergraduate Texts in Mathematics |publisher=Springer |isbn=0-387-90972-9 |page=[https://archive.org/details/generaltopology0000dixm/page/6 6] |quote=According to this definition, an {{em|open neighborhood of $x$}} is nothing more than an open subset of $E$ that contains $x.$}} (resp. closed neighbourhood, compact neighbourhood, etc.). Some authors{{sfn|Engelking|1989|p=12}} require neighbourhoods to be open, so it is important to note their conventions.

File:Neighborhood illust2.svg

A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points. A closed rectangle, as illustrated in the figure, is not a neighbourhood of all its points; points on the edges or corners of the rectangle are not contained in any open set that is contained within the rectangle.

The collection of all neighbourhoods of a point is called the neighbourhood system at the point.

If $S$ is a subset of a topological space $X$, then a neighbourhood of $S$ is a set $V$ that includes an open set $U$ containing $S$,$$S\; \backslash subseteq\; U\; \backslash subseteq\; V\; \backslash subseteq\; X.$$It follows that a set $V$ is a neighbourhood of $S$ if and only if it is a neighbourhood of all the points in $S.$ Furthermore, $V$ is a neighbourhood of $S$ if and only if $S$ is a subset of the interior of $V.$

A neighbourhood of $S$ that is also an open subset of $X$ is called an {{visible anchor|open neighbourhood}} of $S.$

The neighbourhood of a point is just a special case of this definition.

File:Neighborhood illust3.svg

File:Epsilon Umgebung.svg

In a metric space $M\; =\; (X,\; d),$ a set $V$ is a neighbourhood of a point $p$ if there exists an open ball with center $p$ and radius $r>0,$ such that

is contained in $V.$

$V$ is called a uniform neighbourhood of a set $S$ if there exists a positive number $r$ such that for all elements $p$ of $S,$

is contained in $V.$

Under the same condition, for $r\; >\; 0,$ the $r$-neighbourhood $S\_r$ of a set $S$ is the set of all points in $X$ that are at distance less than $r$ from $S$ (or equivalently, $S\_r$ is the union of all the open balls of radius $r$ that are centered at a point in $S$): $$S\_r\; =\; \backslash bigcup\backslash limits\_\{p\backslash in\{\}S\}\; B\_r(p).$$

It directly follows that an $r$-neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an $r$-neighbourhood for some value of $r.$

File:Set of real numbers with epsilon-neighbourhood.svg

Given the set of real numbers $\backslash R$ with the usual Euclidean metric and a subset $V$ defined as

$$V\; :=\; \backslash bigcup\_\{n\; \backslash in\; \backslash N\}\; B\backslash left(n\backslash ,;\backslash ,1/n\; \backslash right),$$

then $V$ is a neighbourhood for the set $\backslash N$ of natural numbers, but is {{em|not}} a uniform neighbourhood of this set.

{{See also|Filters in topology|Topological space#Neighborhood definition|Axiomatic foundations of topological spaces#Definition via neighbourhoods}}

The above definition is useful if the notion of open set is already defined. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points.

A neighbourhood system on $X$ is the assignment of a filter $N(x)$ of subsets of $X$ to each $x$ in $X,$ such that

1. the point $x$ is an element of each $U$ in $N(x)$
2. each $U$ in $N(x)$ contains some $V$ in $N(x)$ such that for each $y$ in $V,$ $U$ is in $N(y).$

One can show that both definitions are compatible, that is, the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system.

In a uniform space $S\; =\; (X,\; \backslash Phi),$ $V$ is called a uniform neighbourhood of $P$ if there exists an entourage $U\; \backslash in\; \backslash Phi$ such that $V$ contains all points of $X$ that are $U$-close to some point of $P;$ that is, $U[x]\; \backslash subseteq\; V$ for all $x\; \backslash in\; P.$

A deleted neighbourhood of a point $p$ (sometimes called a punctured neighbourhood) is a neighbourhood of $p,$ without $\backslash \{p\backslash \}.$ For instance, the interval is a neighbourhood of $p\; =\; 0$ in the real line, so the set $(-1,\; 0)\; \backslash cup\; (0,\; 1)\; =\; (-1,\; 1)\; \backslash setminus\; \backslash \{0\backslash \}$ is a deleted neighbourhood of $0.$ A deleted neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definition of the limit of a function and in the definition of limit points (among other things).{{Cite web |last=Peters |first=Charles |date=2022 |title=Professor Charles Peters |url=https://www.math.uh.edu/~charles/Functions.pdf |access-date=3 April 2022 |website=University of Houston Math}}

• {{annotated link|Isolated point}}
• {{annotated link|Neighbourhood system}}
• {{annotated link|Region (mathematics)}}
• {{annotated link|Tubular neighborhood|Tubular neighbourhood}}

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Category:General topology

Category:Mathematical analysis