creating continual functions from evenly continual functions

Let $(X,d),(Y,d)$ be gloss rooms and also $Y$ compact. Take into consideration the rooms $C_{u}(X,Y)$ of evenly continual functions, and also $C(X,Y)$ of continual functions. We can grant these with the consistent - statistics.

Can $C(X,Y)$ be created from $C_{u}(X,Y)$ making use of just factor - sensible restrictions of $\omega$ - series? I.e. Define $\mathcal{B}_{0}:=C_{u}(X,Y)$, and also $\mathcal{B}_{\alpha}:=$ the set of factor - sensible restrictions of a series $(f_{n})_{n\in\omega}$ of functions $f_{n}\in\bigcup_{\xi<\alpha}\mathcal{B}_{\xi}$. After that this inquiry asks whether $C(X,Y)\subseteq\mathcal{B}_{\omega_{1}}$.

It appears feasible.

Certainly if it holds true, after that we get that all borel functions from $X$ to $Y$ can be created from the evenly continual functions making use of just factor - sensible restrictions.