neighborhood system on a set
In pointset topology, a neighborhood system is defined as the set of neighborhoods^{} of some point in the topological space^{}.
However, one can start out with the definition of a βabstract neighborhood systemβ $\mathrm{\pi \x9d\x94\x91}$ on an arbitrary set $X$ and define a topology^{} $T$ on $X$ based on this system $\mathrm{\pi \x9d\x94\x91}$ so that $\mathrm{\pi \x9d\x94\x91}$ is the neighborhood system of $T$. This is done as follows:
Let $X$ be a set and $\mathrm{\pi \x9d\x94\x91}$ be a subset of $X\Gamma \x97P\beta \x81\u2019(X)$, where $P\beta \x81\u2019(X)$ is the power set^{} of $X$. Then $\mathrm{\pi \x9d\x94\x91}$ is said to be a abstract neighborhood system of $X$ if the following conditions are satisfied:
 1.
if $(x,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$, then $x\beta \x88\x88U$,
 2.
for every $x\beta \x88\x88X$, there is a $U\beta \x8a\x86X$ such that $(x,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$,
 3.
if $(x,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$ and $U\beta \x8a\x86V\beta \x8a\x86X$, then $(x,V)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$,
 4.
if $(x,U),(x,V)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$, then $(x,U\beta \x88\copyright V)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$,
 5.
if $(x,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$, then there is a $V\beta \x8a\x86X$ such that
 β
$(x,V)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$, and
 β
$(y,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$ for all $y\beta \x88\x88V$.
In addition^{}, given this $\mathrm{\pi \x9d\x94\x91}$, define the abstract neighborhood system around $x\beta \x88\x88X$ to be the subset ${\mathrm{\pi \x9d\x94\x91}}_{x}$ of $\mathrm{\pi \x9d\x94\x91}$ consisting of all those elements whose first coordinate is $x$. Evidently, $\mathrm{\pi \x9d\x94\x91}$ is the disjoint union^{} of ${\mathrm{\pi \x9d\x94\x91}}_{x}$ for all $x\beta \x88\x88X$. Finally, let
$T$ $=$ $\mathrm{\{}U\beta \x8a\x86X\beta \x88\pounds \text{for every\Beta}x\beta \x88\x88U\text{,\Beta}(x,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}\}$ $=$ $\mathrm{\{}U\beta \x8a\x86X\beta \x88\pounds \text{for every\Beta}x\beta \x88\x88U\text{, there is a\Beta}V\beta \x8a\x86U\text{, such that\Beta}(x,V)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}\}.$
The two definitions are the same by condition 3. We assert that $T$ defined above is a topology on $X$. Furthermore, ${T}_{x}:=\{U\beta \x88\pounds (x,U)\beta \x88\x88{\mathrm{\pi \x9d\x94\x91}}_{x}\}$ is the set of neighborhoods of $x$ under $T$.
Proof.
We first show that $T$ is a topology. For every $x\beta \x88\x88X$, some $U\beta \x8a\x86X$, we have $(x,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$ by condition 2. Hence $(x,X)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$ by condition 3. So $X\beta \x88\x88T$. Also, $\mathrm{\beta \x88\x85}\beta \x88\x88T$ is vacuously satisfied, for no $x\beta \x88\x88\mathrm{\beta \x88\x85}$. If $U,V\beta \x88\x88T$, then $U\beta \x88\copyright V\beta \x88\x88T$ by condition 4. Let $\{{U}_{i}\}$ be a subset of $T$ whose elements are indexed by $I$ ($i\beta \x88\x88I$). Let $U=\beta \x8b\x83{U}_{i}$. Pick any $x\beta \x88\x88U$, then $x\beta \x88\x88{U}_{i}$ for some $i\beta \x88\x88I$. Since ${U}_{i}\beta \x88\x88T$, $(x,{U}_{i})\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$. Since ${U}_{i}\beta \x8a\x86U$, $(x,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$ by condition 3, so $U\beta \x88\x88T$.
Next, suppose $\mathrm{\pi \x9d\x92\copyright}$ is the set of neighborhoods of $x$ under $T$. We need to show $\mathrm{\pi \x9d\x92\copyright}={T}_{x}$:

1.
($\mathrm{\pi \x9d\x92\copyright}\beta \x8a\x86{T}_{x}$). If $N\beta \x88\x88\mathrm{\pi \x9d\x92\copyright}$, then there is $U\beta \x88\x88T$ with $x\beta \x88\x88U\beta \x8a\x86N$. But $(x,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$, so by condition 3, $(x,N)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$, or $(x,N)\beta \x88\x88{\mathrm{\pi \x9d\x94\x91}}_{x}$, or $N\beta \x88\x88{T}_{x}$.

2.
(${T}_{x}\beta \x8a\x86\mathrm{\pi \x9d\x92\copyright}$). Pick any $U\beta \x88\x88{T}_{x}$ and set $W=\{z\beta \x88\pounds U\beta \x88\x88{T}_{z}\}$. Then $x\beta \x88\x88W\beta \x8a\x86U$ by condition 1. We show $W$ is open. This means we need to find, for each $z\beta \x88\x88W$, a $V\beta \x8a\x86W$ such that $(z,V)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$. If $z\beta \x88\x88W$, then $(z,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$. By condition 5, there is $V\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$ such that $(z,V)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$, and for any $y\beta \x88\x88V$, $(y,U)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$, or $y\beta \x88\x88U$ by condition 1. So $y\beta \x88\x88W$ by the definition of $W$, or $V\beta \x8a\x86W$. Thus $W$ is open and $U\beta \x88\x88\mathrm{\pi \x9d\x92\copyright}$.
This completes^{} the proof. By the way, $W$ defined above is none other than the interior of $U$: $W={U}^{\beta \x88\x98}$. β
Remark. Conversely, if $T$ is a topology on $X$, we can define ${\mathrm{\pi \x9d\x94\x91}}_{x}$ to be the set consisting of $(x,U)$ such that $U$ is a neighborhood of $x$. The the union $\mathrm{\pi \x9d\x94\x91}$ of ${\mathrm{\pi \x9d\x94\x91}}_{x}$ for each $x\beta \x88\x88X$ satisfies conditions $1$ through $5$ above:

1.
(condition 1): clear

2.
(condition 2): because $(x,X)\beta \x88\x88\mathrm{\pi \x9d\x94\x91}$ for each $x\beta \x88\x88X$

3.
(condition 3): if $U$ is a neighborhood of $x$ and $V$ a supserset of $U$, then $V$ is also a neighborhood of $x$

4.
(condition 4): if $U$ and $V$ are neighborhoods of $x$, there are open $A,B$ with $x\beta \x88\x88A\beta \x8a\x86U$ and $x\beta \x88\x88B\beta \x8a\x86V$, so $x\beta \x88\x88A\beta \x88\copyright B\beta \x8a\x86U\beta \x88\copyright V$, which means $U\beta \x88\copyright V$ is a neighborhood of $x$

5.
(condition 5): if $U$ is a neighborhood of $x$, there is open $A$ with $x\beta \x88\x88A\beta \x8a\x86U$; clearly $A$ is a neighborhood of $x$ and any $y\beta \x88\x88A$ has $U$ as neighborhood.
So the definition of a neighborhood system on an arbitrary set gives an alternative way of defining a topology on the set. There is a onetoone correspondence between the set of topologies on a set and the set of abstract neighborhood systems on the set.
Title  neighborhood system on a set 

Canonical name  NeighborhoodSystemOnASet 
Date of creation  20130322 16:41:34 
Last modified on  20130322 16:41:34 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 5400 
Defines  abstract neighborhood system 