Theorem
The real number line setr and the plane setr^2 are not homeomorphic.
Proof
Ifis homeomorphic to
then there is a continuous, one to one and onto function
and
is also continuous, one to one and onto.
Ifis open in setr then
has to be open in
and
is closed so
is closed since
is open.
However with the normal topologies onand
taken as the projection of
onto the real so that
which is closed in
we have
which is neither open nor closed so
and
are not homeomorphic.