Proof That a Union of Path Connected Subsets of a Space With Non Empty Intersection is Path Connected
Letbe a class of non empty connected subsets ofwith non empty intersection so that
Thenis path connected.
LetPath connected subsets ofexist such that
The non empty intersectioncontains at least one point, say
Thenso there is a pathinfromtowithandso there is a pathinfromtowith
Thenis a path fromtoandis connected.