Editing Complex projective space (section)

===Zariski topology===
{{main|Zariski topology}}
In [[algebraic geometry]], complex projective space can be equipped with another topology known as the [[Zariski topology]] {{harv|Hartshorne|1977|loc=§II.2}}.  Let {{nowrap|''S'' {{=}} '''C'''[''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>]}} denote the [[commutative ring]] of polynomials in the (''n''+1) variables ''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>.  This ring is [[graded ring|graded]] by the total degree of each polynomial:
:<math>S = \bigoplus_{n=0}^\infty S_n.</math>
Define a subset of '''CP'''<sup>''n''</sup> to be ''closed'' if it is the simultaneous solution set of a collection of homogeneous polynomials.  Declaring the complements of the closed sets to be open, this defines a topology (the Zariski topology) on '''CP'''<sup>''n''</sup>.