Proof That Each Element of a Topological Space is Contained in Some Component
Each element of a topological spaceis contained in some component of
Obviously each element ofis contained in at least one connected subset of
Letbe the family of all connected subsets ofcontaining
is connected for each
is connected and contains
This union is also unique, since every connected subset ofcontainingis included and non are excluded.
Henceis a maximally connected subset ofcontainingso must be a component of