Why closed points of a variety are Zariski dense?

Question

I was told that closed points are Zariski dense in the variety by my teacher. But I don't understand what does it mean. Every single point in a variety is closed in Zariski topology because it's the zero locus of a maximal ideal. So how could I understand this statement?

Answer 1

It's not true that "every single point in a variety is closed in the Zariski topology". Look at the variety $\mathbb{A}^1 = \operatorname{Spec} \Bbbk[x]$. The generic point corresponding to the zero ideal $(0) \subset \Bbbk[x]$ is not closed. However the set of all closed points of $\mathbb{A}^1$ is dense for the Zariski topology (meaning that its closure is all of $\mathbb{A}^1$). See e.g. this question for a reference (an algebraic variety is in particular of finite type).