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
If
is homeomorphic to
then there is a continuous, one to one and onto function
and
is also continuous, one to one and onto.
If
is open in setr then
has to be open inand
is closed so
is closed sinceis open.
However with the normal topologies onand
taken as the projection ofonto the real so that
which is closed inwe have
which is neither open nor closed soandare not homeomorphic.
