# Is connected component open?

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.

• Think about $\mathbb{R} \setminus \{0\}$. What are the connected components? – Carl Mummert Mar 6 at 2:12
• In fact, it is true in general that any connected component is clopen (it is both open and closed). If you want a proof, you should add your definition of a "connected component" first. – stressed out Mar 6 at 13:41
• sets are not doors, they can be open and closed at the same time... – Henno Brandsma Mar 6 at 17:36