As the title says, I'm looking for a counterexample to the statement that every topological space X is homeomorphic to the disjoint union of its connected components.
I know that this is in fact true if X is locally connected because this implies that the connected components are open, but unfortunately, I don't have a sufficient repertoire of examples of spaces that meet certain conditions (in this case, not being locally connected).
Thanks in advance
J.Dillinger