Connectedness

Abstract

Connectedness seems to promise no surprises; after all it is an inherently pointfree question, the question of non-trivial complemented elements. And the basics are indeed just as in spaces (Section 2). But when we go further, the ways part. We have facts that hold in locales similarly as in spaces but for different reasons (Section 3), and facts that are quite different; of those we present an example in Section 4; so far we have avoided lengthy tedious counterexamples, but in this section, just for once, we present a weird (co)product in detail: besides showing that connectedness is not behaving as one would expect, it also indicates why the fact from Section 3, although parallel to the classical one, cannot be proved by imitating the classical procedure.