Component of Locally Connected Space is Open

Theorem
Let $X$ be a locally connected topological space.

Then its connected components are open.

Proof
By definition of locally connected, $X$ has a basis of connected sets.

Thus $X$ is a union of open connected sets.

By Components are Open iff Union of Open Connected Sets, the connected components of $X$ 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