# Components of a Space

A connected subspace of a topological space is said to be the component of if it is not properly contained in any connected subspace of .

**Note:** The component of a connected space is the whole space, , itself.

**Example:**

The singleton subset of a two-point discrete space is its components.

Let is be two-point discrete space, then the only possible connected subsets of are its singleton subsets. As no singleton subset is properly contained in any other singleton subset, so these singleton subsets are components of . Hence the singleton subsets of a two-point discrete space are its components.

**Theorems:**

• Let be a topological space, then for each , there is exactly one component of containing .

• Let be a topological space, then each connected subset of is contained in a component of .

**Theorem:**

Let be a topological space, then every component is closed in .

**Proof:**

Let be a component of . If possible, suppose that is not closed in . Since is closed and connected and contains , which is a contradiction to the fact that is a component. Hence , so is closed.