CZ-Groups and the Zariski Topology

Abstract

Let U be the space of n-row vectors over the field F and R = F [X₁,..., X _n ], the polynomial ring over F in n indeterminates. A subset A of U is said to be closed in U if there exists a subset S of R such that A is the set of zeros of S, that is if

$$ A = \left\{ {\left( {{a_{1}}, \ldots ,{a_{n}}} \right) \in U:f\left( {{a_{1}}, \ldots ,{a_{n}}} \right) = 0\;for\;all\;f\;in\;S} \right\} $$

If S is any subset of R let V(S) denote the set of zeros of S (in U). Note that

$$ V\left( S \right) = V\left( {ideal\;generated\;by\;S} \right) $$

and

$$ \mathop{ \cap }\limits_{\alpha } V\left( {{S_{\alpha }}} \right) = V\left( {\mathop{ \cup }\limits_{\alpha } {S_{\alpha }}} \right) $$

.