# Topology/Bases

Topology
 ← Topological Spaces Bases Points in Sets →

## DefinitionEdit

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 BaseEdit

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 $x\in B\subseteq U$.

## Constructing Topologies from BasesEdit

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 $X$.

## SemibasesEdit

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 $\mathcal{S}$.

## ExercisesEdit

1. Show that the collection $\mathcal{B}=\{(a,b):a,b\in\mathbb{R},a 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 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 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 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; 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 $X\,\!$.

Topology
 ← Topological Spaces Bases Points in Sets →