16

Connectedness

**16.1 Connectedness**

In Section 5.3 of Volume I we introduced the notion of connectedness of subsets of the real line, and showed that a non-empty subset of **R** is connected if and only if it is an interval. The notion extends easily to topological spaces. A topological space *splits* if *X* = *F*_{1} ∪ *F*_{2}, where *F*_{1} and *F*_{2} are disjoint nonempty closed subsets of (*X*, *τ*). The decomposition *X* = *F*_{1} ∪ *F*_{2} is a *splitting* of *X*. If *X* does not split, it is *connected*. A subset *A* of (*X*, *τ*) is connected if it is connected as a topological subspace of (*X*, *τ*). If *X* = *F*_{1} ∪ *F*_{2} is a splitting, then *F*_{1} = *C*(*F*_{2}) and *F*_{2} = *C*(*F*_{1}) are open sets, and so *X* is the disjoint union of two non-empty sets which are both open and closed; conversely if *U* is a non-empty proper ...

Start Free Trial

No credit card required