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
Best Answer
Two standard examples of totally disconnected spaces (that are not discrete) are $\mathbb{Q}$ (with the subspace topology from $\mathbb{R}$) and the product $\{ 0, 1 \}^{\mathbb{N}}$ of countably many copies of the two-point discrete space (homemorphic to both the Cantor set and, if I recall correctly, the $p$-adic integers).
Generalizing the second example, any Stone space is totally disconnected. These spaces are important as they arise as the spectra of Boolean rings.
If you want to expand your repertoire of examples of spaces, it's hard to do better than Steen and Seebach's Counterexamples in Topology.