question on discrete variety

Question

Please help me to solve this question Or introduce references that help me.

Suppose $X$ is a discrete variety. Show that $X$ is finite.

Thank you.

Answer 1

I assume you mean by discrete variety that $X$ carries the discrete topology.

The underlying topological space of a variety is noetherian, i.e. every descending chain of closed subsets stabilizes at some point (cf. Hartshorne, Algebraic Geometry p.5 for the definition of noetherian).

Now you can show that a noetherian topological space, which is also Hausdorff, must be a finite, discrete space. (cf. ibid., Ex.I.1.7). This result covers your problem as a special case, because the discrete topology is Hausdorff.

Answer 2

Alternatively, you can use that a noetherian topological space has only finitely many irreducible components.