Zariski Topology

Definition (Zariski Topology)

In algebraic topology, we use the Zariski Topology where the open sets are the complements of the zero-sets (sets where the polynomial vanishes) of the polynomial function.

In particular, when $X=\mathbb{R}$, the open sets are just the complements of the finite subsets of $\mathbb{R}$. This is also called cofinite topology on $\mathbb{R}$.

Lemma (Arbitrary Unions/Intersections of Cofinite Sets)

Let $X$ be infinite, and $A,B$ be finite. Then $(X-A)$ and $(X-B)$ are cofinite. Then,

$$\begin{array}{rl}(X-A)\cup (X-B)& =X-(A\cap B)\\ (X-A)\cap (B-B)& =X-(A\cup B)\end{array}$$