Neighbourhood system

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In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and related areas of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter $\mathcal\left(x\right)$ for a point $x$ in a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is the collection of all
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
s of $x.$

# Definitions

Neighbourhood of a point or set An of a point (or
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
) $x$ in a topological space $X$ is any
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
$U$ of $X$ that contains $x.$ A is any subset $N \subseteq X$ that contains open neighbourhood of $x$; explicitly, $N$ is a neighbourhood of $x$ in $X$
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
there exists some open subset $U$ with $x \in U \subseteq N$. Equivalently, a neighborhood of $x$ is any set that contains $x$ in its
topological interior In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the ...
. Importantly, a "neighbourhood" does have to be an open set; those neighbourhoods that also happen to be open sets are known as "open neighbourhoods." Similarly, a neighbourhood that is also a
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
(respectively,
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
,
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
, etc.) set is called a (respectively, , , etc.). There are many other types of neighbourhoods that are used in topology and related fields like
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. The family of all neighbourhoods having a certain "useful" property often forms a
neighbourhood basis In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x. Definitions Neighbour ...
, although many times, these neighbourhoods are not necessarily open.
Locally compact space In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
s, for example, are those spaces that, at every point, have a neighbourhood basis consisting entirely of compact sets. Neighbourhood filter The neighbourhood system for a point (or
non-empty In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other t ...
subset) $x$ is a
filter Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component tha ...
called the The neighbourhood filter for a point $x \in X$ is the same as the neighbourhood filter of the
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermeloâ€“Fraenkel set theory, the ...
$\.$

## Neighbourhood basis

A or (or or ) for a point $x$ is a
filter base In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then ...
of the neighbourhood filter; this means that it is a subset $\mathcal \subseteq \mathcal(x)$ such that for all $V \in \mathcal\left(x\right),$ there exists some $B \in \mathcal$ such that $B \subseteq V.$ That is, for any neighbourhood $V$ we can find a neighbourhood $B$ in the neighbourhood basis that is contained in $V.$ Equivalently, $\mathcal$ is a local basis at $x$ if and only if the neighbourhood filter $\mathcal$ can be recovered from $\mathcal$ in the sense that the following equality holds: (See Chapter 2, Section 4) $\mathcal(x) = \left\\!\!\;.$ A family $\mathcal \subseteq \mathcal\left(x\right)$ is a neighbourhood basis for $x$ if and only if $\mathcal$ is a
cofinal subset In mathematics, a subset B \subseteq A of a preordered set (A, \leq) is said to be cofinal or frequent in A if for every a \in A, it is possible to find an element b in B that is "larger than a" (explicitly, "larger than a" means a \leq b). Cofin ...
of $\left\left(\mathcal\left(x\right), \supseteq\right\right)$ with respect to the
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
$\supseteq$ (importantly, this partial order is the
superset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
relation and not the
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
relation).

## Neighbourhood subbasis

A at $x$ is a family $\mathcal$ of subsets of $X,$ each of which contains $x,$ such that the collection of all possible finite intersections of elements of $\mathcal$ forms a neighbourhood basis at $x.$

# Examples

If $\R$ has its usual
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...
then the neighborhoods of $0$ are all those subsets $N \subseteq \R$ for which there exists some
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
$r > 0$ such that $\left(-r, r\right) \subseteq N.$ For example, all of the following sets are neighborhoods of $0$ in $\R$: but none of the following sets are neighborhoods of $0$: $\, \; \Q, \; (0,2), \; [0, 2), \; [0, 2) \cup \Q, \; (-2, 2) \setminus \left\$ where $\Q$ denotes the rational numbers. If $U$ is an open subset of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
$X$ then for every $u \in U,$ $U$ is a neighborhood of $u$ in $X.$ More generally, if $N \subseteq X$ is any set and $\operatorname_X N$ denotes the
topological interior In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of the ...
of $N$ in $X,$ then $N$ is a neighborhood (in $X$) of every point $x \in \operatorname_X N$ and moreover, $N$ is a neighborhood of any other point. Said differently, $N$ is a neighborhood of a point $x \in X$ if and only if $x \in \operatorname_X N.$ Neighbourhood bases In any topological space, the neighbourhood system for a point is also a neighbourhood basis for the point. The set of all open neighbourhoods at a point forms a neighbourhood basis at that point. For any point $x$ in a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, the sequence of
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
s around $x$ with radius $1/n$ form a
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
neighbourhood basis $\mathcal = \left\$. This means every metric space is
first-countable In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
. Given a space $X$ with the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
the neighbourhood system for any point $x$ only contains the whole space, $\mathcal\left(x\right) = \$. In the
weak topology In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
on the space of measures on a space $E,$ a neighbourhood base about $\nu$ is given by $\left\$ where $f_i$ are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
bounded functions from $E$ to the real numbers and $r_1, \dots, r_n$ are positive real numbers. Seminormed spaces and topological groups In a
seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
, that is a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
with the
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
induced by a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
, all neighbourhood systems can be constructed by
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
of the neighbourhood system for the origin, $\mathcal(x) = \mathcal(0) + x.$ This is because, by assumption, vector addition is separately continuous in the induced topology. Therefore, the topology is determined by its neighbourhood system at the origin. More generally, this remains true whenever the space is a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
or the topology is defined by a pseudometric.

# Properties

Suppose $u \in U \subseteq X$ and let $\mathcal$ be a neighbourhood basis for $u$ in $X.$ Make $\mathcal$ into a
directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...
by partially ordering it by superset inclusion $\,\supseteq.$ Then $U$ is a neighborhood of $u$ in $X$ if and only if there exists an $\mathcal$-indexed
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
$\left\left(x_N\right\right)_$ in $X \setminus U$ such that $x_N \in N \setminus U$ for every $N \in \mathcal$ (which implies that $\left\left(x_N\right\right)_ \to u$ in $X$).