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Let $(X,d),(Y,d)$ be polish spaces and $Y$ compact. Consider the spaces $C_{u}(X,Y)$ of uniformly continuous functions, and $C(X,Y)$ of continuous functions. We can endow these with the uniform-metric.

Can $C(X,Y)$ be generated from $C_{u}(X,Y)$ using only point-wise limits of $\omega$-sequences? I.e. Define $\mathcal{B}_{0}:=C_{u}(X,Y)$, and $\mathcal{B}_{\alpha}:=$ the set of point-wise limits of a sequence $(f_{n})_{n\in\omega}$ of functions $f_{n}\in\bigcup_{\xi<\alpha}\mathcal{B}_{\xi}$. Then this question asks whether $C(X,Y)\subseteq\mathcal{B}_{\omega_{1}}$.

It seems possible.

Of course if it's true, then we get that all borel functions from $X$ to $Y$ can be generated from the uniformly continuous functions using only point-wise limits.

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