Component of Locally Connected Space is Open

Theorem
Let $T = \left({S, \tau}\right)$ be a locally connected topological space.

Then its connected components are open.

Proof
By definition of locally connected, $T$ has a basis of connected subsets of $S$ in $T$.

Thus $S$ is a union of open connected subsets of $S$ in $T$.

By Components are Open iff Union of Open Connected Sets, the connected components of $T$ are open.

Also see

 * Path Components of Locally Path-Connected Space are Open, an analogous result for path components
 * Locally Connected iff Components of Open Subsets are Open