A topological space X is said to be totally disconnected space if any pair of distinct of X can be separated by a disconnection of X.
            In other words, a topological space X is said to be totally disconnected space if for any two points x and y of X, there is a disconnection  of X such that  and .
            In other words, a topological space X is said to be 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 , then  and  are open subsets of X such that . Since  is a disconnection of X, so X is totally disconnected.

Examples:

• One point space is totally disconnected.
•  is totally disconnected.
•  is totally disconnected.
• The Cantor set is totally disconnected.
•  with usual topology is not totally disconnected.
•  with upper limit topology generated by open-closed intervals  is totally disconnected.

Theorems:

• Every totally disconnected space is 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.