Neighbourhood system

 ( General Topology ) 

In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods,

or neighbourhood filter $\backslash mathcal\{N\}(x)$ for a point $x$ in a topological space is the collection of all neighbourhoods of $x.$

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Definitions Neighbourhood of a point or set

An of a point (or subset) $x$ in a topological space $X$ is any Open set $U$ of $X$ that contains $x.$ A is any subset $N\; \backslash subseteq\; X$ that contains open neighbourhood of $x$; explicitly, $N$ is a neighbourhood of $x$ in $X$ if and only if there exists some open subset $U$ with $x\; \backslash in\; U\; \backslash subseteq\; N$. Equivalently, a neighborhood of $x$ is any set that contains $x$ in its topological interior.

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 set (respectively, compact space, connected space, 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. The family of all neighbourhoods having a certain "useful" property often forms a neighbourhood basis, although many times, these neighbourhoods are not necessarily open. Locally compact spaces, 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 empty set subset) $x$ is a filter called the The neighbourhood filter for a point $x\; \backslash in\; X$ is the same as the neighbourhood filter of the singleton set $\backslash \{x\backslash \}.$

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Neighbourhood basis A or (or or ) for a point $x$ is a filter base of the neighbourhood filter; this means that it is a subset $$\backslash mathcal\{B\}\; \backslash subseteq\; \backslash mathcal\{N\}(x)$$ such that for all $V\; \backslash in\; \backslash mathcal\{N\}(x),$ there exists some $B\; \backslash in\; \backslash mathcal\{B\}$ such that $B\; \backslash subseteq\; V.$ Here, $\backslash mathcal\{N\}(x)$ denotes the set of all neighbourhoods of x. That is, for any neighbourhood $V$ we can find a neighbourhood $B$ in the neighbourhood basis that is contained in $V.$

Equivalently, $\backslash mathcal\{B\}$ is a local basis at $x$ if and only if the neighbourhood filter $\backslash mathcal\{N\}$ can be recovered from $\backslash mathcal\{B\}$ in the sense that the following equality holds:

(See Chapter 2, Section 4) $$\backslash mathcal\{N\}(x)\; =\; \backslash left\backslash \{\; V\; \backslash subseteq\; X\; ~:~\; B\; \backslash subseteq\; V\; \backslash text\{\; for\; some\; \}\; B\; \backslash in\; \backslash mathcal\{B\}\; \backslash right\backslash \}\backslash !\backslash !\backslash ;.$$ A family $\backslash mathcal\{B\}\; \backslash subseteq\; \backslash mathcal\{N\}(x)$ is a neighbourhood basis for $x$ if and only if $\backslash mathcal\{B\}$ is a Cofinal set of $\backslash left(\backslash mathcal\{N\}(x),\; \backslash supseteq\backslash right)$ with respect to the partial order $\backslash supseteq$ (importantly, this partial order is the superset relation and not the subset relation).

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

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Examples If $\backslash R$ has its usual Euclidean topology then the neighborhoods of $0$ are all those subsets $N\; \backslash subseteq\; \backslash R$ for which there exists some real number $r\; >\; 0$ such that $(-r,\; r)\; \backslash subseteq\; N.$ For example, all of the following sets are neighborhoods of $0$ in $\backslash R$: $$(-2,\; 2),\; \backslash ;\; -2,2,\; \backslash ;\; -2,\; \backslash cup\; \backslash Q,\; \backslash ;\; \backslash R$$ but none of the following sets are neighborhoods of $0$: $$\backslash \{0\backslash \},\; \backslash ;\; \backslash Q,\; \backslash ;\; (0,2),\; \backslash ;\; [0,\; 2),\; \backslash ;\; [0,\; 2)\; \backslash cup\; \backslash Q,\; \backslash ;\; (-2,\; 2)\; \backslash setminus\; \backslash left\backslash \{1,\; \backslash tfrac\{1\}\{2\},\; \backslash tfrac\{1\}\{3\},\; \backslash tfrac\{1\}\{4\},\; \backslash ldots\backslash right\backslash \}$$ where $\backslash Q$ denotes the .

If $U$ is an open subset of a topological space $X$ then for every $u\; \backslash in\; U,$ $U$ is a neighborhood of $u$ in $X.$ More generally, if $N\; \backslash subseteq\; X$ is any set and $\backslash operatorname\{int\}\_X\; N$ denotes the topological interior of $N$ in $X,$ then $N$ is a neighborhood (in $X$) of every point $x\; \backslash in\; \backslash operatorname\{int\}\_X\; N$ and moreover, $N$ is a neighborhood of any other point. Said differently, $N$ is a neighborhood of a point $x\; \backslash in\; X$ if and only if $x\; \backslash in\; \backslash operatorname\{int\}\_X\; N.$

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, the sequence of around $x$ with radius $1/n$ form a countable neighbourhood basis $\backslash mathcal\{B\}\; =\; \backslash left\backslash \{B\_\{1/n\}\; :\; n\; =\; 1,2,3,\backslash dots\; \backslash right\backslash \}$. This means every metric space is first-countable.

Given a space $X$ with the indiscrete topology the neighbourhood system for any point $x$ only contains the whole space, $\backslash mathcal\{N\}(x)\; =\; \backslash \{X\backslash \}$.

In the weak topology on the space of measures on a space $E,$ a neighbourhood base about $\backslash nu$ is given by where $f\_i$ are continuous bounded functions from $E$ to the real numbers and $r\_1,\; \backslash dots,\; r\_n$ are positive real numbers.

Seminormed spaces and topological groups

In a seminormed space, that is a vector space with the topology induced by a seminorm, all neighbourhood systems can be constructed by translation of the neighbourhood system for the origin, $$\backslash mathcal\{N\}(x)\; =\; \backslash mathcal\{N\}(0)\; +\; x.$$

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 or the topology is defined by a pseudometric.

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Properties Suppose $u\; \backslash in\; U\; \backslash subseteq\; X$ and let $\backslash mathcal\{N\}$ be a neighbourhood basis for $u$ in $X.$ Make $\backslash mathcal\{N\}$ into a directed set by partial order it by superset inclusion $\backslash ,\backslash supseteq.$ Then $U$ is a neighborhood of $u$ in $X$ if and only if there exists an $\backslash mathcal\{N\}$-indexed net $\backslash left(x\_N\backslash right)\_\{N\; \backslash in\; \backslash mathcal\{N\}\}$ in $X\; \backslash setminus\; U$ such that $x\_N\; \backslash in\; N\; \backslash setminus\; U$ for every $N\; \backslash in\; \backslash mathcal\{N\}$ (which implies that $\backslash left(x\_N\backslash right)\_\{N\; \backslash in\; \backslash mathcal\{N\}\}\; \backslash to\; u$ in $X$).

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See also

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Bibliography

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