How to use the transitivity of homeomorphism to show that a

How to use the transitivity of homeomorphism to show that a disc is homeomorphic to the whole plane, but not to a line or a point?

Solution

Let D denote a disk , P the whole plane , L a line and Q a point.

D is homeomorphic to the unit square U , which in turn is homeomorphic to P, hence D is homeomorphic to P.

If D were homeomorphic to L or Q , then by transitivity, P is homeomorphic to L or Q which is trivially false (by connectedness argument)