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.