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.