Indiscrete Topology

Last updated Nov 1, 2022

Let $X$ be a Set. Then ${\emptyset, X}$ is a topology on $X$. It is called the Indiscrete Topology on $X$.

1. $\emptyset, X \in {\emptyset, X}$ by construction $\checkmark$.
2. Finite Set Intersections either include $\emptyset$ or they don’t. If they do, then the Set Intersection is $\emptyset$, otherwise it is $X$ $\checkmark$.
3. Set Unions either include $X$ or they don’t. If they do, then the Set Union is $X$. Otherwise it is $\emptyset$ $\checkmark$. $\blacksquare$

1. The Indiscrete Topology is the coarsest Topological Space. Every Topological Space on a Set contains the Indiscrete Topology.
2. The Indiscrete Topology is not Hausdorff. It can serve as a useful counterexample when trying to test if a statement holds outside of Hausdorff spaces.