Universal functions and generalized classes of functions

Abstract:

For a class $\mathcal {A}$ of subsets of a set $Z$ which is closed under countable unions we consider the families of functions \[ \underline M \mathcal {A} = \{ f:Z \to [0,1]:(\forall c)({f^{ - 1}}((c,1]) \in \mathcal {A})\} \] and \[ \overline M \mathcal {A} = \{ f:Z \to [0,1]:(\forall c)({f^{ - 1}}([0,c)) \in \mathcal {A})\} \] (for instance, if $Z$ is a topological space and $\mathcal {A}$ is the family of all open subsets of $Z$, then $\underline M \mathcal {A}$ and $\overline M \mathcal {A}$ are the families of lower and upper semicontinuous functions from $Z$ to $[0,1]$, respectively). Using universal functions we show that under certain natural assumptions about $\mathcal {A}$ there exists a function $f \in \underline M \mathcal {A}$ such that there is no partition $\{ {X_n}:n \in {\mathbf {N}}\}$ of $Z$ and a family of functions $\{ {h_n}:n \in {\mathbf {N}}\} \subseteq M\mathcal {A}$ such that $f = { \cup _n}({h_n}|{X_n})$. This is a generalization of some results of this type proved by Novikov and Adian, Keldyš, and Laczkovich for the Baire hierarchy of functions. The universal functions technique we use is different from the methods of these authors.