Discrete Topology

Last updated Nov 1, 2022

Let $X$ be a Set. Then $\mathcal{P}(X)$ is a topology on $X$. It is known as the Discrete Topology.

1. $X, \emptyset$ in $\mathcal{P}(X)$ because they are subsets of $X$.
2. $\mathcal{P}(X)$ is closed under Set Union, because Set Unions of subsets are still subsets Set Union
3. $\mathcal{P}(X)$ is closed under finite Set Intersection because Set Intersection of subsets is still a subset.

$\blacksquare$

1. Every subset of $X$ is Closed. This is because for any $A \subset X$, $A^{C}$ is also a subset of $X$, so $A^{C} \in \mathcal{P}(X)$.
2. By (1), every singleton Set is both Closed and Open, so no Set with more than one element can be Connected. Thus the singletons are the Connected Components and $X$ is Totally Disconnected.
3. The Discrete Topology has a Topological Basis of singletons, $\mathcal{B} = {{x} \in \mathcal{P}(X) : x \in X}$. This is because each singleton is a subset of $X$, so it is Open, and every subset $A \subset X$ can be written as $\bigcup\limits_{x \in A}{x}$.
4. The Discrete Topology is the finest possible Topological Space. It contains every other Topological Space on $X$. This is by definition, since a topology is a subset of $\mathcal{P}(X)$.

• Power Set.