conditions for a collection of subsets to be a basis for some topology
Not just any collection^{} of subsets of $X$ can be a basis for a topology^{} on $X$. For instance, if we took $\mathrm{\pi \x9d\x92\x9e}$ to be all open intervals of length $1$ in $\mathrm{\beta \x84\x9d}$, $\mathrm{\pi \x9d\x92\x9e}$ isnβt the basis for any topology on $\mathrm{\beta \x84\x9d}$: $(0,1)$ and $(.5,1.5)$ are unions of elements of $\mathrm{\pi \x9d\x92\x9e}$, but their intersection^{} $(.5,1)$ is not. The collection formed by arbitrary unions of members of $\mathrm{\pi \x9d\x92\x9e}$ isnβt closed under finite intersections and isnβt a topology.
Weβd like to know which collections $\mathrm{\beta \x84\neg}$ of subsets of $X$ could be the basis for some topology on $X$. Hereβs the result:
Theorem.
A collection $\mathrm{\beta \x84\neg}$ of subsets of $X$ is a basis for some topology on $X$ if and only if:

1.
Every $x\beta \x88\x88X$ is contained in some ${B}_{x}\beta \x88\x88\mathrm{\beta \x84\neg}$, and

2.
If ${B}_{1}$ and ${B}_{2}$ are two elements of $\mathrm{\beta \x84\neg}$ containing $x\beta \x88\x88X$, then thereβs a third element ${B}_{3}$ of $\mathrm{\beta \x84\neg}$ such that $x\beta \x88\x88{B}_{3}\beta \x8a\x82{B}_{1}\beta \x88\copyright {B}_{2}$.
Proof.
First, weβll show that if $\mathrm{\beta \x84\neg}$ is the basis for some topology $\mathrm{\pi \x9d\x92\u2015}$ on $X$, then it satisfies the two conditions listed.
$\mathrm{\pi \x9d\x92\u2015}$ is a topology on $X$, so $X\beta \x88\x88\mathrm{\pi \x9d\x92\u2015}$. Since $\mathrm{\beta \x84\neg}$ is a basis for $\mathrm{\pi \x9d\x92\u2015}$, that means $X$ can be written as a union of members of $\mathrm{\beta \x84\neg}$: since every $x\beta \x88\x88X$ is in this union, every $x\beta \x88\x88X$ is contained in some member of $\mathrm{\beta \x84\neg}$. That takes care of the first condition.
For the second condition: if ${B}_{1}$ and ${B}_{2}$ are elements of $\mathrm{\beta \x84\neg}$, theyβre also in $\mathrm{\pi \x9d\x92\u2015}$. $\mathrm{\pi \x9d\x92\u2015}$ is closed under intersection, so ${B}_{1}\beta \x88\copyright {B}_{2}$ is open in $\mathrm{\pi \x9d\x92\u2015}$. Then ${B}_{1}\beta \x88\copyright {B}_{2}$ can be written as a union of members of $\mathrm{\beta \x84\neg}$, and any $x\beta \x88\x88{B}_{1}\beta \x88\copyright {B}_{2}$ is contained by some basis element in this union.
Second, weβll show that if a collection $\mathrm{\beta \x84\neg}$ of subsets of $X$ satisfies the two conditions, then the collection $\mathrm{\pi \x9d\x92\u2015}$ of unions of members of $\mathrm{\beta \x84\neg}$ is a topology on $X$.

β’
$\mathrm{\beta \x88\x85}\beta \x88\x88\mathrm{\pi \x9d\x92\u2015}$: $\mathrm{\beta \x88\x85}$ is the null union of zero elements of $\mathrm{\beta \x84\neg}$.

β’
$X\beta \x88\x88\mathrm{\pi \x9d\x92\u2015}$: by the first condition, every $X$ is contained in some member of $\mathrm{\beta \x84\neg}$. The union of all the members of $\mathrm{\beta \x84\neg}$ is then all of $X$.

β’
$\mathrm{\pi \x9d\x92\u2015}$ is closed under arbitrary unions: Say we have a union of sets ${T}_{\mathrm{\Xi \pm}}\beta \x88\x88\mathrm{\pi \x9d\x92\u2015}$β¦
$\underset{\mathrm{\Xi \pm}\beta \x88\x88I}{\beta \x8b\x83}}{T}_{\mathrm{\Xi \pm}$ $={\displaystyle \underset{\mathrm{\Xi \pm}\beta \x88\x88I}{\beta \x8b\x83}}{\displaystyle \underset{\mathrm{\Xi \xb2}\beta \x88\x88{J}_{\mathrm{\Xi \pm}}}{\beta \x8b\x83}}{B}_{\mathrm{\Xi \xb2}}$ (since each ${T}_{\mathrm{\Xi \pm}}$ is a union of sets in $\mathrm{\beta \x84\neg}$) $={\displaystyle \underset{\mathrm{\Xi \xb2}\beta \x88\x88{\beta \x8b\x83}_{\mathrm{\Xi \pm}\beta \x88\x88I}{J}_{\mathrm{\Xi \pm}}}{\beta \x8b\x83}}{B}_{\mathrm{\Xi \xb2}}$ Since thatβs a union of elements of $\mathrm{\beta \x84\neg}$, itβs also a member of $\mathrm{\pi \x9d\x92\u2015}$.

β’
$\mathrm{\pi \x9d\x92\u2015}$ is closed under finite intersections: since a collection of sets is closed under finite intersections if and only if it is closed under pairwise intersections, we need only check that the intersection of two members ${T}_{1},{T}_{2}$ of $\mathrm{\pi \x9d\x92\u2015}$ is in $\mathrm{\pi \x9d\x92\u2015}$.
Any $x\beta \x88\x88{T}_{1}\beta \x88\copyright {T}_{2}$ is contained in some ${B}_{x}^{1}\beta \x8a\x82{T}_{1}$ and ${B}_{x}^{2}\beta \x8a\x82{T}_{2}$. By the second condition, $x\beta \x88\x88{B}_{x}^{1}\beta \x88\copyright {B}_{x}^{2}$ gets us a ${B}_{x}^{3}$ with $x\beta \x88\x88{B}_{x}^{3}\beta \x8a\x82{B}_{x}^{1}\beta \x88\copyright {B}_{x}^{2}\beta \x8a\x82{T}_{1}\beta \x88\copyright {T}_{2}$. Then
$${T}_{1}\beta \x88\copyright {T}_{2}=\underset{x\beta \x88\x88{T}_{1}\beta \x88\copyright {T}_{2}}{\beta \x8b\x83}{B}_{x}^{3}$$ which is in $\mathrm{\pi \x9d\x92\u2015}$.
β
Title  conditions for a collection of subsets to be a basis for some topology 

Canonical name  ConditionsForACollectionOfSubsetsToBeABasisForSomeTopology 
Date of creation  20130322 14:21:49 
Last modified on  20130322 14:21:49 
Owner  waj (4416) 
Last modified by  waj (4416) 
Numerical id  4 
Author  waj (4416) 
Entry type  Proof 
Classification  msc 54A99 
Classification  msc 54D70 