A class of spaces that admit to
a special kind of mapping onto the nonnegative real numbers is considered and it is
shown that this particular type of space is invariant under compact monotone
mappings. It is also shown that if such a space admits to a one to one (monotone)
mapping onto a “nice” subset of the plane then the mapping is a homeomorphism
(compact monotone mapping). If the one to one mapping (monotone) is not a
homeomorphism (compact monotone mapping) then its range necessarily separates
E2.
