There is a theorem that:A space is locally connected iff each connected components of an open set is open.
But recently I had seen to prove That each connected component is closed. Connected Components are Closed
Then how can the connected component of an open set be open if it is a locally connected space ? It will be contradiction to the statement that connected components is closed.
Best Answer
A subset being closed doesn't preclude that subset from being open. For a simple example, every discrete space is locally connected, and every subset of a discrete space—in particular the singleton sets (which are the connected components)—is both open and closed.