Totally Disconnected Space

A topological space $X$ is said to be a totally disconnected space if any distinct pair of $X$ can be separated by a disconnection of $X$.

In other words, a topological space $X$ is said to be a totally disconnected space if for any two points $x$ and $y$ of $X$, there is a disconnection $\left\{ {A,B} \right\}$ of $X$ such that $x \in A$ and $y \in B$.

Or, a topological space $X$ is said to be a totally disconnected space if its connected subsets are only the singleton subset of $X$.

 

Example:

Every discrete space is totally disconnected.

Let $X$ be a discrete space. Let $x,y \in X,{\text{ }}x \ne y$, then $A = \left\{ x \right\}$ and ${A^c} = X\backslash \left\{ x \right\}$ are open subsets of $X$ such that $x \in A,{\text{ y}} \in {A^c}$. Since $\left\{ {A,{A^c}} \right\}$ is a disconnection of $X$, then $X$ is totally disconnected.

Examples:
• One point in space is totally disconnected.
• $\mathbb{Q} \subseteq \mathbb{R}$ is totally disconnected.
• ${\mathbb{Q}^c} \subseteq \mathbb{R}$ is totally disconnected.
• The Cantor set is totally disconnected.
• $\mathbb{R}$ with usual topology is not totally disconnected.
• $\mathbb{R}$ with upper limit topology generated by open-closed intervals $\left( {a,b} \right]$ is totally disconnected.

Theorems
• Every totally disconnected space is a Hausdorff space.
• The components of a totally disconnected space are its singleton subsets.
• If a Hausdorff space $X$ has an open base whose sets are also closed then $X$ is totally disconnected.