# Totally Disconnected Space

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

In other words, a topological space is said to be a totally disconnected space if for any two points and of , there is a disconnection of such that and .

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

**Example:**

Every discrete space is totally disconnected.

Let be a discrete space. Let , then and are open subsets of such that . Since is a disconnection of , then is totally disconnected.

**Examples:**

• One point in 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 a Hausdorff space.

• The components of a totally disconnected space are its singleton subsets.

• If a Hausdorff space has an open base whose sets are also closed then is totally disconnected.

Saqib Ali

June 23@ 2:41 pmI need proof of this theorem