Editing Base (topology) (section)

==Base for the closed sets==

[[Closed set]]s are equally adept at describing the topology of a space. There is, therefore, a dual notion of a base for the closed sets of a topological space. Given a topological space <math>X,</math> a [[Family of sets|family]] <math>\mathcal{C}</math> of closed sets forms a '''base for the closed sets''' if and only if for each closed set <math>A</math> and each point <math>x</math> not in <math>A</math> there exists an element of <math>\mathcal{C}</math> containing <math>A</math> but not containing <math>x.</math>
A family <math>\mathcal{C}</math> is a base for the closed sets of <math>X</math> if and only if its {{em|dual}} in <math>X,</math> that is the family <math>\{X\setminus C: C\in \mathcal{C}\}</math> of [[Complement (set theory)|complements]] of members of <math>\mathcal{C}</math>, is a base for the open sets of <math>X.</math>

Let <math>\mathcal{C}</math> be a base for the closed sets of <math>X.</math> Then
#<math>\bigcap \mathcal{C} = \varnothing</math>
#For each <math>C_1, C_2 \in \mathcal{C}</math> the union <math>C_1 \cup C_2</math> is the intersection of some subfamily of <math>\mathcal{C}</math> (that is, for any <math>x \in X</math> not in <math>C_1 \text{ or } C_2</math> there is some <math>C_3 \in \mathcal{C}</math> containing <math>C_1 \cup C_2</math> and not containing <math>x</math>).
Any collection of subsets of a set <math>X</math> satisfying these properties forms a base for the closed sets of a topology on <math>X.</math> The closed sets of this topology are precisely the intersections of members of <math>\mathcal{C}.</math>

In some cases it is more convenient to use a base for the closed sets rather than the open ones. For example, a space is [[completely regular]] if and only if the [[zero set]]s form a base for the closed sets. Given any topological space <math>X,</math> the zero sets form the base for the closed sets of some topology on <math>X.</math> This topology will be the finest completely regular topology on <math>X</math> coarser than the original one. In a similar vein, the [[Zariski topology]] on '''A'''<sup>''n''</sup> is defined by taking the zero sets of polynomial functions as a base for the closed sets.