Discrete topology

Example

Let $X$ be a set and let $\tau = \mathcal{P}(X)$ be the set of all subsets of $X$. Then $\tau$ is a topology on $X$, the discrete topology. The space $(X, \tau)$ is a discrete space.

Remarks

Each subset of $X$ is both open and closed. Each point $x$ gives an open set $\{x\}$. It is the finest topology on $X$. It is the metric topology of the discrete metric. Each function from a discrete space to any topological space is continuous. A discrete space is compact if and only if it is finite.

Explanations

L1 human

In the discrete topology, every set is open, so each point is isolated from all other points. The integers $\mathbb{Z}$, as a subspace of $\mathbb{R}$, carry the discrete topology.