Counterexample of a disconnected topological space with open components but not locally connected

Question

Theorem 25.3 in Munkres proved that a space is locally connected if the components of every open subset is open.

What obviously follows from the theorem is that the components of a locally connected space is open. However, is there a nontrivial counterexample (in this case, disconnected space) where the components are open but the space is not locally connected?

Answer 1

Of course. Pick $X\sqcup \{*\}$ for any connected but not locally connected $X$.