>>13165281No problemo. Things are easy when there are finitely many components. Each of them is always closed, so any union of them is also closed. This means that you can exclude any single component and its complement will be closed, and so all components are clopen. This doesn't work in general, though, as you may have infinitely many components and the unions no longer need all be closed. If you have a locally connected space, then that will already give you open components (prove it!), and you will have a lot of guaranteed clopens; the components and their finite unions.
