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 $$\mathcal{S}=\{[a,b):a,b\in\mathbb{R},a0\,\!$$, 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 $$X\,\!$$.