# Geometric Picture of Open Subsets of Irreducible spaces in Zariski topology

I understand why nonempty open subsets of irreducible topological spaces are dense. I am having trouble comprehending how this translates to Zariski's topology. We know that a finite affine variety is irreducible iff it is a single point. How do you take an open subset of a single point? What does this look like geometrically?

• The statement "an affine variety is irreducible iff it is a single point" is not true. How did you come to that? Feb 4, 2023 at 19:56
• @KReiser Andreas Gathmann Chapter 2; example 2.11: "A finite affine variety is irreducible if and only if it is connected: namely if and only if it contains at most one point." Am I interpreting this wrong? Feb 4, 2023 at 20:00
• @SassatelliGiulio ^ Feb 4, 2023 at 20:00
• Lots of varieties are not finite! For instance, $\Bbb A^1$. Feb 4, 2023 at 20:01
• @KReiser You are right. I mean my question with respect to finite affine varieties. So not $\mathbb{A}^n$ Feb 4, 2023 at 20:05