An arbitrary Hausdorff compactification of a Tychonoff space X obtained from a -base by a modified Wallman method

Let ( Z , h ) be an arbitrary Hausdorff compactification of a Tychonoff space X, D = { f | f = f ○ ○ h , f ○ ∈ C ( Z ) } and let B D be the collection of all finite unions of all nonempty finite intersections of elements of C = { f −1 ( [ a , b ] ) | a , b ∈ R , a < b , f ∈ D and f −1 ( [ a , b ] ) ≠ ∅ } . Then B D is a base for the closed sets of X, and is called a C D ∗ -base for X induced by D. This paper will show that ( Z , h ) can be obtained from the C D ∗ -base B D for X by a modified Wallman method.