# general topology – Disconnected subspace

We know that if $$X$$ is a disconnected topological space and $$C$$ is a connected component of $$X$$, then $$C$$ is closed. Besides, if $$X$$ is locally connected, then $$C$$ is open and closed. Now suppose $$Ysubset X$$ is a disconnected (topological) subspace of $$X$$. The proof of the above results suggest that each connected component $$C$$ of $$Y$$ is closed in $$X$$ (not only in $$Y$$), and if $$X$$ is locally connected, then $$C$$ is open in $$X$$ (not only open in $$Y$$). Am I right?