Universal functions and generalized classes of functions
Jacek Cichoń, Michał Morayne · Proceedings of the American Mathematical Society · 1988
For a class A \mathcal {A} of subsets of a set Z Z which is closed under countable unions we consider the families of functions \[ M _ A = { f : Z → [ 0 , 1 ] : ( ∀ c ) ( f − 1 ( ( c , 1 ] ) ∈ A ) } \underline M \mathcal {A} = \{ f:Z \to [0,1]:(\forall c)({f^{ - 1}}((c,1]) \in \mathcal {A})\} \] and \[ M ¯ A = { f : Z → [ 0 , 1 ] : ( ∀ c ) ( f − 1 ( [ 0 , c ) ) ∈ A ) } \overline M \mathcal {A} = \{ f:Z \to [0,1]:(\forall c)({f^{ - 1}}([0,c)) \in \mathcal {A})\} \] (for instance, if Z Z is a topological space and