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.