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We have the following. Fix $X, (Y,d)$ polish spaces where $d$ is some bounded metricon $Y$. Assign the topology to . Topologise $Z=C^{0}(X,Y)$ induced C^{0}(X,Y)$ by the metric $d(f,g)=sup_{x\in X}d(f(x),g(x))$. One Then one can tweak Kechris' proof to show, that the subspace $S$ of uniformly continuous maps with bounded images, is Polish. Is it possible to show that $C^{0}(X,Y)$ can be generated by $S$, using point-wise limits of $\omega$-sequences of functions? This would be a useful result. |
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We have the following. Fix $X, (Y,d)$ polish spaces where $d$ is some bounded metric on $Y$. Assign the topology to $Z=C^{0}(X,Y)$ induced by the metric $d(f,g)=sup_{x\in X}d(f(x),g(x))$. One can tweak Kechris' proof to show that the subspace of uniformly continuous maps with bounded images, is Polish. |
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