Topology/Bases

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[edit] 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.

[edit] 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 $x\in B\subseteq U$ .

[edit] 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:

If $x\in X$ , then there exists a $B\in\mathcal{B}$ such that $x\in B$ .
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 $X$ .

[edit] Semibases

Let $X$ be any set and $\mathcal{S}$ 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.

[edit] Exercises

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 $\mathbb{R}$ .
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 $\mathbb{R}$ .
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 $\mathbb{R}$ .
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 $\mathbb{R}$ .
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 $X\,\!$ .