Proof That a Connected Set Cannot be Expressed as the Union of Nonempty, Disjoint, Closed Subsets

Theorem

A connected set cannot be expressed as the union of nonempty, disjoint, closed subsets.

Proof

Supposeis connected and closed setsexist with

Thenand

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.