Let $X$ be a nonempty set. Let d be the discrete metric on $X$ ($d(x, y) = 1$ for distinct $x, y ∈ X$). Show that every $E ⊂ X$ is open.

Question

Let $X$ be a nonempty set. Let d be the discrete metric on $X$ ($d(x, y) = 1$ for distinct $x, y ∈ X$). Show that every $E ⊂ X$ is open.

I am learning metric spaces and I didn't understand some things:

Every finite subset $E\subset X$ is closed.

In case $E$ is a finite subset , why $E$ is open ? Is it possible that $X$ is a finite set , then every $E ⊂ X$ is finite.

I think I have a misunderstanding about the definitions and I'll be grateful for getting a clarifiction.

Thanks ! (and sorry for my bad english).