The absolute (EX,πX) is
constructed for an arbitrary space X and is shown to be unique with respect to EX
being extremally disconnected and completely regular and πX being a 𝜃-continuous,
perfect, separating irreducible surjection. A function f : X → Y is said to have a
continuous E-lifting if there is a continuous function F : EX → EY such that
πY∘ F = f ∘ πX. A class of functions, called η-continuous, is introduced, shown to
contain the class of continuous functions and the class of 𝜃-continuous, closed
surjections, and proved to have continuous E-liftings. Functions which have
continuous E-liftings are completely characterized as being the composition of
η-continuous functions.
