Vol. 94, No. 2, 1981

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Completely regular absolutes and projective objects

Raymond Frank Dickman, Jack Ray Porter and Leonard Rubin

Vol. 94 (1981), No. 2, 277–295

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.

Mathematical Subject Classification 2000
Primary: 54G05
Secondary: 54C05
Received: 8 February 1980
Revised: 29 May 1980
Published: 1 June 1981
Raymond Frank Dickman
Jack Ray Porter
Leonard Rubin