Proof That Any Open Connected Subset of Rn is Polygonally Connected
Ifis an open connected subset ofthenis polygonally connected.
Supposeis open and connected and
Letbe all points ofwhich can be polygonally connected to
Sinceis open for anythere is an open neighbourhoodofsuch thatcan be polygonally connected to any point ofhencehenceis open.
The setconsists of elements which cannot be polygonally connected tosois
whereandare disjoint open subsets ofSinceis connected, we have either
but sincesoandis polygonally connected.