# Components of a Space

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

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

**Example:**

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

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

**Theorems**

• Let $$X$$ be a topological space, then for each $$x \in X$$ there is exactly one component of $$X$$ containing $$x$$.

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

**Theorem**

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

**Proof**

Let $$C$$ be a component of $$X$$. If possible, suppose that $$C$$ is not closed in $$X$$. Since $$\overline C $$ is closed and connected and contains $$C$$, this is a contradiction to the fact that $$C$$ is a component. Hence $$C = \overline C $$, so $$C$$ is closed.