## Proof That the Components of a Space Partition the Space

Theorem

Letbe the set of components of a spaceThe components ofform a partition ofi.e.andif

Proof

Any elementbelongs to at least one connected subspace i.e. at least one of the All theare connected. Iffor anythen x would have to be in some other connected subspaceBut then- a contradiction. Hencefor some

Take two setsandSupposeandThenis a connected subset ofcontaining bothand- a contradiction sinceif