# Basis for a topological space

This article is about a basic definition in topology.VIEW: Definitions built on this | Facts about this | Survey articles about this

View a complete list of basic definitions in topology

## Definition when the topological space is specified

### Symbol-free definition

A **basis for a topological space** is a collection of open subsets of the topological space, such that every open subset can be expressed as a (possibly empty) union of basis subsets.

### Definition with symbols

If is a topological space, a **basis** for is a collection of open subsets of (here, is an indexing set) such that for any open subset of , there exists such that:

## Definition when the topological space is not specified

### Symbol-free definition

Given a set, a collection of subsets of the set is said to form a **basis for a topological space** or a **basis for a topology** if the following two conditions are satisfied:

- The union of all members of the collection is the whole space
- Any finite intersection of members of the collection, is itself a union of members of the collection

The topology *generated* by this basis is the topology in which the open sets are precisely the unions of basis sets.

### Definition with symbols

In symbols: if is a set, a collection of subsets of is said to form a **basis for a topology** on if the following two conditions are satisfied:

- For all , there exists such that:

The second condition is sometimes stated as follows: if , then there exists such that .

The topology generated by the s is defined as follows: a subset is open in if and only if there exists

### Equivalence of definitions

`Further information: Equivalence of definitions of basis`

## Related notions

## Examples

### Extreme examples

- For any topological space, the collection of
*all*open subsets is a basis. That's because any open subset of a topological space can be expressed as a union of size one. - For a discrete topological space, the collection of one-point subsets forms a basis. That's because every open subset of a discrete topological space is a union of one-point subsets, namely, the one-point subsets corresponding to its elements.

### Examples from metric spaces

- Euclidean space: A basis for the usual topology on Euclidean space is the open balls. An open ball of radius centered at a point , is defined as the set of all whose distance from is strictly smaller than . By the way the topology on is defined, these open balls clearly form a basis.
- Metric space: Given any metric space, there is a natural way of viewing it as a topological space. This natural way involves declaring the collection of open balls in the metric space as a basis. To see that this gives a well-defined topology, we need to check that the collection of open balls satisfies the conditions to be a basis for a topological space. This follows from the conditions for a metric space, including nonnegativity, symmetry and the triangle inequality.