Topology/Bases

Definition

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

Obviously $𝒯$ is a base for itself.

Conditions for Being a Base

In a topological space $(X,𝒯)$ a collection $ℬ$ is a base for $𝒯$ 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 $B\in ℬ$ 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 $B\in ℬ$. However, the if part is obvious, from the facts that the elements in $B\in ℬ$ 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 $ℬ$, 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 $B_x\in ℬ$. The union of all of them indeed is $U$. Thus, any open set can be formed as a union of sets within $ℬ$.

Constructing Topologies from Bases

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

1. If $x\in X$, then there exists a $B\in ℬ$ such that $x\in B$.
2. If $B_1,B_2\in ℬ$ and $x\in B_1\cap B_2$, then there is a $B\in ℬ$ 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 $ℬ$ is $X$.

Semibases

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

Exercises

1. Show that the collection $ℬ=\{(a,b):a,b\in ℝ,a<b\}$ of all open intervals in $ℝ$ is a base for a topology on $ℝ$.
2. Show that the collection $𝒞=\{[a,b]:a,b\in ℝ,a<b\}$ of all closed intervals in $ℝ$ is not a base for a topology on $ℝ$.
3. Show that the collection $ℒ=\{(a,b]:a,b\in ℝ,a<b\}$ of half open intervals is a base for a topology on $ℝ$.
4. Show that the collection $𝒮=\{[a,b):a,b\in ℝ,a<b\}$ of half open intervals is a base for a topology on $ℝ$.
5. Let $a,b\in ℝ$. A Partition $𝒫$ over the closed interval $[a,b]$ is defined as the ordered n-tuple $a<x_1<x_2<\dots <x_n<b$; the norm of a partition $𝒫$ is defined as $\Vert 𝒫\Vert =\sup \{x_i-x_{i-1}|2\le i\le n\}$
   For every $\delta >0$, define the set $U_{\delta }=\{𝒫|\Vert 𝒫\Vert \le \delta \}$.
   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 $X$.

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