Totally disconnected space that is not $T_2$

Question

The Wikipedia article on totally disconnected spaces seems to imply they are not necessarily Hausdorff (they are all $T_1$ though). What's an example of a totally disconnected non $T_2$ space?

(A space is totally disconnected if all connected components are singletons.)

Answer 1

Take $X = \mathbb N \cup \{ - \infty , + \infty \}$ with the topology where

• each point of $\mathbb N$ is isolated,
• each neighborhood of $- \infty$ and $+ \infty$ is a cofinite subset of $X$ (containing the respective point).

Note that for each $n \in \mathbb N$ the singleton $\{ n \}$ is clopen in $X$.

As $- \infty , + \infty$ cannot be separated by disjoint open sets, the space is not Hausdorff.

Suppose $A \subseteq X$ contains at least two points.

• If $A$ contains a point of $\mathbb N$, then for any $n \in A \cap \mathbb N$ the sets $\{ n \}$ and $X \setminus \{ n \}$ witness that $A$ is disconnected.
• Otherwise $A = \{ -\infty , + \infty \}$, in which case $X \setminus \{ + \infty \} , X \setminus \{ - \infty \}$ witness that $A$ is disconnected.