Proof That Elements of a Connected Space Can be Joined By a Simple Chain Consisting of Elements of the Open Cover of the Space
Theorem
Letbe a connected set and letbe an open cover ofAny two points ofcan be joined by a simple chain consisting of elements of.
Proof
Subsetsofare said to form a simple chain between pointsandif

Onlycontains

Onlycontains

if
Letand letbe the set of all points ofwhich are to be joined toby a simple chain consisting of elements of
Letthen there exists a simple chainfromtoAll elements of belong tosince the chainconnects x with any element ofhence
Sinceis open, so is
Suppose now thatSinceis a cover ofthere is an elementsuch that
Supposethen there is some chain fromtoand a contradiction henceand
Henceis open andis closed.
We concudeand the theorem is proved.