# Topology/Bases

From Wikibooks, open books for an open world

< Topology

## Contents

## Definition[edit]

Let be a topological space. A collection of open sets is called a **base** for the topology if every open set is the union of sets in .

Obviously is a base for itself.

## Conditions for Being a Base[edit]

In a topological space a collection is a base for if and only if it consists of open sets and for each point and open neighborhood of there is a set such that .

## Constructing Topologies from Bases[edit]

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

- If , then there exists a such that .
- If and , then there is a such that .

**Remark :** Note that the first condition is equivalent to saying that *The union of all sets in is .*

## Semibases[edit]

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

## Exercises[edit]

- Show that the collection of all open intervals in is a base for a topology on .
- Show that the collection of all closed intervals in is
**not**a base for a topology on . - Show that the collection of half open intervals is a base for a topology on .
- Show that the collection of half open intervals is a base for a topology on .
- Let . A
**Partition**over the closed interval is defined as the ordered n-tuple ; the**norm**of a partition is defined as

For every , define the set .

If is the set of all partitions on , prove that the collection of all is a Base over the Topology on .