Proof That a Connected Set Cannot be Expressed as the Union of Nonempty, Disjoint, Closed Subsets
A connected set cannot be expressed as the union of nonempty, disjoint, closed subsets.
Supposeis connected and closed setsexist with
The complement of a closed set is open, so thatandare also both open and is the union of open setsandHenceis not connected.
Conversely supposeis disconnected. Then nonempty open setsexist such thaand
Thenandso thatandare closed - a contradiction.