Editing Base (topology) (section)

==Examples==

The set {{math|Γ}} of all open intervals in <math>\mathbb{R}</math> forms a basis for the [[Euclidean topology]] on <math>\mathbb{R}</math>.

A non-empty family of subsets of a set {{mvar|X}} that is closed under finite intersections of two or more sets, which is called a [[Pi-system|{{pi}}-system]] on {{mvar|X}}, is necessarily a base for a topology on {{mvar|X}} if and only if it covers {{mvar|X}}.  By definition, every [[Sigma-algebra|σ-algebra]], every [[Filter (set theory)|filter]] (and so in particular, every [[Neighborhood system|neighborhood filter]]), and every [[Topological space#topology|topology]] is a covering {{pi}}-system and so also a base for a topology.  In fact, if {{math|Γ}} is a filter on {{mvar|X}} then {{math|{ ∅ } ∪ Γ}} is a topology on {{mvar|X}} and {{math|Γ}} is a basis for it.  A base for a topology does not have to be closed under finite intersections and many are not.  But nevertheless, many topologies are defined by bases that are also closed under finite intersections.  For example, each of the following families of subset of {{mvar|<math>\mathbb{R}</math>}} is closed under finite intersections and so each forms a basis for ''some'' topology on <math>\mathbb{R}</math>:
* The set {{math|Γ}} of all <em>bounded</em> open intervals in <math>\mathbb{R}</math> generates the usual [[Euclidean topology]] on <math>\mathbb{R}</math>. 
* The set {{math|Σ}} of all bounded <em>closed</em> intervals in <math>\mathbb{R}</math> generates the [[discrete topology]] on <math>\mathbb{R}</math> and so the Euclidean topology is a subset of this topology. This is despite the fact that {{math|Γ}} is not a subset of {{math|Σ}}. Consequently, the topology generated by {{math|Γ}}, which is the [[Euclidean topology]] on <math>\mathbb{R}</math>, is [[Comparison of topologies|coarser than]] the topology generated by {{math|Σ}}. In fact, it is <em>strictly</em> coarser because {{math|Σ}} contains non-empty compact sets which are never open in the Euclidean topology. 
* The set {{math|Γ<sub><math>\mathbb{Q}</math></sub>}} of all intervals in {{math|Γ}} such that both endpoints of the interval are [[rational number]]s generates the same topology as {{math|Γ}}. This remains true if each instance of the symbol {{math|Γ}} is replaced by {{math|Σ}}.
* {{math|1=Σ<sub>∞</sub> = { [''r'', ∞) : ''r'' ∈ <math>\mathbb{R}</math> } }} generates a topology that is [[Comparison of topologies|strictly coarser]] than the topology generated by {{math|Σ}}. No element of {{math|1=Σ<sub>∞</sub>}} is open in the Euclidean topology on <math>\mathbb{R}</math>.
* {{math|1=Γ<sub>∞</sub> = { (''r'', ∞) : ''r'' ∈ <math>\mathbb{R}</math> } }} generates a topology that is strictly coarser than both the [[Euclidean topology]] and the topology generated by {{math|Σ<sub>∞</sub>}}. The sets {{math|Σ<sub>∞</sub>}} and {{math|Γ<sub>∞</sub>}} are disjoint, but nevertheless {{math|Γ<sub>∞</sub>}} is a subset of the topology generated by {{math|Σ<sub>∞</sub>}}.

===Objects defined in terms of bases===

* The [[order topology]] on a totally ordered set admits a collection of open-interval-like sets as a base.
* In a [[metric space]] the collection of all [[open ball]]s forms a base for the topology.
* The [[discrete topology]] has the collection of all [[singleton (mathematics)|singleton]]s as a base.
* A [[second-countable space]] is one that has a [[countable]] base.

The [[Zariski topology]] on the [[spectrum of a ring]] has a base consisting of open sets that have specific useful properties. For the usual base for this topology, every finite intersection of basic open sets is a basic open set.

* The [[Zariski topology]] of <math>\C^n</math> is the topology that has the [[algebraic set]]s as closed sets. It has a base formed by the [[set complement]]s of [[affine algebraic hypersurface|algebraic hypersurface]]s. 
* The Zariski topology of the [[spectrum of a ring]] (the set of the [[prime ideals]]) has a base such that each element consists of all prime ideals that do not contain a given element of the ring.