Editing Base (topology) (section)

==Definition and basic properties==

Given a [[topological space]] <math>(X,\tau)</math>, a '''base'''{{sfnm|1a1 = Willard|1y = 2004|1loc = Definition 5.1|2a1 = Engelking|2y = 1989|2p = 12|3a1 = Bourbaki|3y = 1989|3loc = Definition 6, p. 21|4a1 = Arkhangel'skii|4a2 = Ponomarev|4y = 1984|4p = 40}} (or '''basis'''{{sfn|Dugundji|1966|loc = Definition 2.1, p. 64}}) for the [[topology (structure)|topology]] <math>\tau</math> (also called a ''base for'' <math>X</math> if the topology is understood) is a [[Family of sets|family]] <math>\mathcal{B}\subseteq\tau</math> of open sets such that every open set of the topology can be represented as the union of some subfamily of <math>\mathcal{B}</math>.<ref group=note>The [[empty set]], which is always open, is the union of the empty family.</ref>  The elements of <math>\mathcal{B}</math> are called ''basic open sets''.
Equivalently, a family <math>\mathcal{B}</math> of subsets of <math>X</math> is a base for the topology <math>\tau</math> if and only if <math>\mathcal{B}\subseteq\tau</math> and for every open set <math>U</math> in <math>X</math> and point <math>x\in U</math> there is some basic open set <math>B\in\mathcal{B}</math> such that <math>x\in B\subseteq U</math>.

For example, the collection of all [[open interval]]s in the [[real line]] forms a base for the standard topology on the real numbers.  More generally, in a metric space <math>M</math> the collection of all open balls about points of <math>M</math> forms a base for the topology.

In general, a topological space <math>(X,\tau)</math> can have many bases.  The whole topology <math>\tau</math> is always a base for itself (that is, <math>\tau</math> is a base for <math>\tau</math>).  For the real line, the collection of all open intervals is a base for the topology.  So is the collection of all open intervals with rational endpoints, or the collection of all open intervals with irrational endpoints, for example.  Note that two different bases need not have any basic open set in common.  One of the [[topological properties]] of a space <math>X</math> is the minimum [[cardinality]] of a base for its topology, called the '''weight''' of <math>X</math> and denoted <math>w(X)</math>.  From the examples above, the real line has countable weight.

If <math>\mathcal{B}</math> is a base for the topology <math>\tau</math> of a space <math>X</math>, it satisfies the following properties:{{sfnm|1a1 = Willard|1y = 2004|1loc = Theorem 5.3|2a1 = Engelking|2y = 1989|2p = 12}}
:(B1) The elements of <math>\mathcal{B}</math> ''[[cover (topology)|cover]]'' <math>X</math>, i.e., every point <math>x\in X</math> belongs to some element of <math>\mathcal{B}</math>.
:(B2) For every <math>B_1,B_2\in\mathcal{B}</math> and every point <math>x\in B_1\cap B_2</math>, there exists some <math>B_3\in\mathcal{B}</math> such that <math>x\in B_3\subseteq B_1\cap B_2</math>.
Property (B1) corresponds to the fact that <math>X</math> is an open set; property (B2) corresponds to the fact that <math>B_1\cap B_2</math> is an open set.

Conversely, suppose <math>X</math> is just a set without any topology and <math>\mathcal{B}</math> is a family of subsets of <math>X</math> satisfying properties (B1) and (B2).  Then <math>\mathcal{B}</math> is a base for the topology that it generates.  More precisely, let <math>\tau</math> be the family of all subsets of <math>X</math> that are unions of subfamilies of <math>\mathcal{B}.</math>  Then <math>\tau</math> is a topology on <math>X</math> and <math>\mathcal{B}</math> is a base for <math>\tau</math>.{{sfnm|1a1 = Willard|1y = 2004|1loc = Theorem 5.3|2a1 = Engelking|2y = 1989|2loc = Proposition 1.2.1}}
(Sketch: <math>\tau</math> defines a topology because it is stable under arbitrary unions by construction, it is stable under finite intersections by (B2), it contains <math>X</math> by (B1), and it contains the empty set as the union of the empty subfamily of <math>\mathcal{B}</math>.  The family <math>\mathcal{B}</math> is then a base for <math>\tau</math> by construction.)  Such families of sets are a very common way of defining a topology.

In general, if <math>X</math> is a set and <math>\mathcal{B}</math> is an arbitrary collection of subsets of <math>X</math>, there is a (unique) smallest topology <math>\tau</math> on <math>X</math> containing <math>\mathcal{B}</math>.  (This topology is the [[intersection (set theory)|intersection]] of all topologies on <math>X</math> containing <math>\mathcal{B}</math>.)  The topology <math>\tau</math> is called the '''topology generated by''' <math>\mathcal{B}</math>, and <math>\mathcal{B}</math> is called a [[subbase]] for <math>\tau</math>.  
The topology <math>\tau</math> consists of <math>X</math> together with all arbitrary unions of finite intersections of elements of <math>\mathcal{B}</math> (see the article about [[subbase]].)  Now, if <math>\mathcal{B}</math> also satisfies properties (B1) and (B2), the topology generated by <math>\mathcal{B}</math> can be described in a simpler way without having to take intersections: <math>\tau</math> is the set of all unions of elements of <math>\mathcal{B}</math> (and <math>\mathcal{B}</math> is a base for <math>\tau</math> in that case).

There is often an easy way to check condition (B2).  If the intersection of any two elements of <math>\mathcal{B}</math> is itself an element of <math>\mathcal{B}</math> or is empty, then condition (B2) is automatically satisfied (by taking <math>B_3=B_1\cap B_2</math>).  For example, the [[Euclidean topology]] on the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set.  But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.

An example of a collection of open sets that is not a base is the set <math>S</math> of all semi-infinite intervals of the forms <math>(-\infty,a)</math> and <math>(a,\infty)</math> with <math>a\in\mathbb{R}</math>.  The topology generated by <math>S</math> contains all open intervals <math>(a,b)=(-\infty,b)\cap(a,\infty)</math>, hence <math>S</math> generates the standard topology on the real line.  But <math>S</math> is only a subbase for the topology, not a base: a finite open interval <math>(a,b)</math> does not contain any element of <math>S</math> (equivalently, property (B2) does not hold).