Topology/Bases

Definition

Let $(X,{\mathcal {T}})$ be a topological space. A collection ${\mathcal {B}}$ of open sets is called a base for the topology ${\mathcal {T}}$ if every open set $U$ is the union of sets in ${\mathcal {B}}$.

Obviously ${\mathcal {T}}$ is a base for itself.

Conditions for Being a Base

In a topological space $(X,{\mathcal {T}})$ a collection ${\mathcal {B}}$ is a base for ${\mathcal {T}}$ if and only if it consists of open sets and for each point $x\in X$ and open neighborhood $U$ of $x$ there is a set $B\in {\mathcal {B}}$ such that $x\in B\subseteq U$.

Proof:
We need to show that a subset $U$ of $X$ is open if and only if it is a union of elements in $B\in {\mathcal {B}}$. However, the if part is obvious, from the facts that the elements in $B\in {\mathcal {B}}$ are open, and that so are arbitrary unions of open sets. Thus, we only have to prove, that any open set $U$ indeed is such a union.
Let $U$ be any open set. Consider any element $x\in U$. By assumption, there is at least one element in ${\mathcal {B}}$, which both contains $x$ and is a subset of $U$. By the axiom of choice, we may simultaneously for each $x\in U$ choose such an element $B_{x}\in {\mathcal {B}}$. The union of all of them indeed is $U$. Thus, any open set can be formed as a union of sets within ${\mathcal {B}}$.

Constructing Topologies from Bases

Let $X$ be any set and ${\mathcal {B}}$ a collection of subsets of $X$. There exists a topology ${\mathcal {T}}$ on $X$ such that ${\mathcal {B}}$ is a base for ${\mathcal {T}}$ if and only if ${\mathcal {B}}$ satisfies the following:

1. If $x\in X$, then there exists a $B\in {\mathcal {B}}$ such that $x\in B$.
2. If $B_{1},B_{2}\in {\mathcal {B}}$ and $x\in B_{1}\cap B_{2}$, then there is a $B\in {\mathcal {B}}$ such that $x\in B\subseteq B_{1}\cap B_{2}$.

Remark : Note that the first condition is equivalent to saying that The union of all sets in ${\mathcal {B}}$ is $X$.

Semibases

Let $X$ be any set and ${\mathcal {S}}$ a collection of subsets of $X$. Then ${\mathcal {S}}$ is a semibase if a base of X can be formed by a finite intersection of elements of ${\mathcal {S}}$.

Exercises

1. Show that the collection ${\mathcal {B}}=\{(a,b):a,b\in \mathbb {R} ,a<b\}$ of all open intervals in $\mathbb {R}$ is a base for a topology on $\mathbb {R}$.
2. Show that the collection ${\mathcal {C}}=\{[a,b]:a,b\in \mathbb {R} ,a<b\}$ of all closed intervals in $\mathbb {R}$ is not a base for a topology on $\mathbb {R}$.
3. Show that the collection ${\mathcal {L}}=\{(a,b]:a,b\in \mathbb {R} ,a<b\}$ of half open intervals is a base for a topology on $\mathbb {R}$.
4. Show that the collection ${\mathcal {S}}=\{[a,b):a,b\in \mathbb {R} ,a<b\}$ of half open intervals is a base for a topology on $\mathbb {R}$.
5. Let $a,b\in \mathbb {R}$. A Partition ${\mathcal {P}}$ over the closed interval $[a,b]\,\!$ is defined as the ordered n-tuple $a<x_{1}<x_{2}<\ldots <x_{n}<b\,\!$; the norm of a partition ${\mathcal {P}}$ is defined as $\|{\mathcal {P}}\|=\sup\{x_{i}-x_{i-1}|2\leq i\leq n\}$
   For every $\delta >0\,\!$, define the set $U_{\delta }=\{{\mathcal {P}}|\|{\mathcal {P}}\|\leq \delta \}$.
   If $X\,\!$ is the set of all partitions on $[a,b]\,\!$, prove that the collection of all $U_{\delta }\,\!$ is a Base over the Topology on $X\,\!$.