# Topology/Bases

 Topology ← Topological Spaces Bases Points in Sets →

## Definition

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

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

## Conditions for Being a Base

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

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

## Constructing Topologies from Bases

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

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

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

## Semibases

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

## Exercises

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

 Topology ← Topological Spaces Bases Points in Sets →