Question

Determine the closed subsets of a.set X with the discrete metric

Answer 1

Let $x\in X$. Consider the open ball $B$ of radius $r$ where $r<1$. Clearly, $B$ is equal to the singleton set $\{x\}$.

So $x\in B$ and $B$ is an improper subset of $\{x\}$. Hence the set $\{x\}$ is open. This implies that every singleton set is open, which implies that every subset of $X$ is open. So if $Y \sub X$ then $Y^c$ is open. Hence $Y$ is closed. Thus every subset of a discrete space is closed