Path Component of Locally Path-Connected Space is Open

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

Then its path components are open.

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

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

By Path Components are Open iff Union of Open Path-Connected Sets, the path components of $X$ are open.

Also see

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