Discrete topology

Definition: Discrete, indiscrete topology

Given any set $X$, the collection of all subsets of $X$ (i.e., the power set $\mathcal{P}(X)$) is a topology on $X$ called the discrete topology. The collection $\{\varnothing ,X\}$ is also a topology called the indiscrete or trivial topology.

Note that a topological space is discrete (i.e., every set is an open set) if and only if every singleton (sets containing a single element) is also open.

Definition: Discrete metric

For any set $X$, the metric generating the discrete topology is given by

$$d(x,y)=\{\begin{array}{r}1x\ne y\\ 0x=y\end{array}.$$

Recall that to show a topology is the discrete topology, it suffices to show every singleton is open, and we have $\{x\}=B_d(x,1/2)$ for all $x\in X$.