View source for Base (topology)

A '''basis''' or '''base''' of a [[topology]] is a collection <math>\mathcal{B}</math> of sets in a topological space <math>(X,\tau)</math> that classify the set of [[open set]]s of the space. 

#For any basis <math>\mathcal{B}</math>, the union of the sets in <math>\mathcal{B}</math> is equal to <math>X</math>. Phrased differently, for any element <math>x \in X</math>, there exists a basis set <math>A \in \mathcal{B}</math> such that <math>x \in A</math>. 
#For any two sets <math>A, B \in \mathcal{B}</math>, given an element <math>x \in A \cap B</math>, then there exists another set <math>C \in \mathcal{B}</math> such that <math>x \in C \subset A \cap B</math>. 

This definition is very useful for comparing different topologies. In particular, we have the following theorem:

''Theorem'': In a space <math>X</math>, given two topologies <math>\tau</math> and <math>\tau'</math>, then <math>\tau \subset \tau'</math> [[iff]] for any basis element <math>B \in \mathcal{B}</math> and any element <math>x \in B</math>, there exists a basis element <math>B' \in \mathcal{B}'</math> such that <math>x \in B' \subset B</math>.  

== Sub-basis ==
A '''sub-basis''' <math>\mathcal{S}</math> of <math>X</math> is a collection of sets whose union is <math>X</math>. The collection of intersection of sets in <math>\mathcal{S}</math> forms a basis <math>\mathcal{B}</math> on <math>X</math>. 

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[[Category:Topology]]