X is never homeomorphic to R X is homeomorphic to R if and only if X is countable X is homeomorphic to R if and only if X is infinite X is homeomorphic to R if and only if X is finite We define the...

Extracted text: X is never homeomorphic to R X is homeomorphic to R if and only if X is countable X is homeomorphic to R if and only if X is infinite X is homeomorphic to R if and only if X is finite We define the included point topology by Tp={ UCR;U=Ø or pEU}. Let A = [3,5[, then A is dense in R if * R is equipped with the usual topology R is equipped with Tp and p = 5 None of the choices O Ris equipped with Tp and p =3 Ris not connected if T is * the trivial (usual) A ɔgy the finite closed topology