Proof That the Line and the Plane Are Not Homeomorphic

Theorem

The real number line setr and the plane setr^2 are not homeomorphic.

Proof

Ifis homeomorphic tothen there is a continuous, one to one and onto functionandis also continuous, one to one and onto.

Ifis open in setr thenhas to be open inandis closed sois closed sinceis open.

However with the normal topologies onandtaken as the projection ofonto the real so thatwhich is closed inwe havewhich is neither open nor closed soandare not homeomorphic.