Is the space of continuous functions from a compact metric space into a Polish space Polish? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T16:05:41Z http://mathoverflow.net/feeds/question/46011 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46011/is-the-space-of-continuous-functions-from-a-compact-metric-space-into-a-polish-sp Is the space of continuous functions from a compact metric space into a Polish space Polish? Byron Schmuland 2010-11-14T03:50:59Z 2010-12-06T04:01:09Z <p>Let $K$ be a compact metric space, and $(E,d_E)$ a complete separable metric space. Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with the metric $d(f,g)=\sup_{k\in K}\ d_E (f(k),g(k))$. Is the space $C$ separable?</p> <p>The result is true when $E$ is the real line; this is Corollary 11.2.5 in Dudley's book <em>Real Analysis and Probability</em>. </p> <p>The result is also true when $K=[0,1]$ (if I'm not being too careless) by considering $C$ as a subspace of the Skorohod space $D_E[0,1]$, which is complete and separable by Theorem 5.6 in Ethier and Kurtz's book <em>Markov Processes: Characterization and Convergence</em>. </p> <p>For general $K$, it is not so obvious how to find an explicit countable dense set in $C$, but I suspect one could modify Ethier and Kurtz's approach and get a proof. </p> <p>But surely this result is known, and stated in some book? I've searched through my library without success. </p> <hr> <p><strong>Update:</strong> This result is also <strong>Theorem 2.4.3</strong> of S. M. Srivastava's book <em>A Course on Borel Sets</em>. His proof is the same as Kechris's. I have also found an alternative, but false, published proof using the "fact" that $C(K,E)$ is $\sigma$-compact. Beware! </p> http://mathoverflow.net/questions/46011/is-the-space-of-continuous-functions-from-a-compact-metric-space-into-a-polish-sp/46015#46015 Answer by Ed Dean for Is the space of continuous functions from a compact metric space into a Polish space Polish? Ed Dean 2010-11-14T04:17:51Z 2010-11-14T04:17:51Z <p>Yes, it appears e.g. as Theorem 4.19 in Chapter I of Kechris' <em>Classical Descriptive Set Theory</em>. (The relevant page is visible in Google Books if it's not in your library.)</p> http://mathoverflow.net/questions/46011/is-the-space-of-continuous-functions-from-a-compact-metric-space-into-a-polish-sp/48418#48418 Answer by Raj for Is the space of continuous functions from a compact metric space into a Polish space Polish? Raj 2010-12-06T03:51:09Z 2010-12-06T04:01:09Z <p>We have the following. Fix $X, (Y,d)$ polish spaces where $d$ is some bounded metric. Topologise $C^{0}(X,Y)$ by the metric $d(f,g)=sup_{x\in X}d(f(x),g(x))$. Then one can tweak Kechris' proof to show, that the subspace $S$ of uniformly continuous maps with bounded images, is Polish.</p> <p>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.</p>