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As in title,when a function space with compact-open topology has countable chain condition? Are there some sufficient and necessary conditions? Who give some references about this topic?

McCoy and Ntantu [Topological Properties of Spaces of Continuous Functions, Page 68] pointed out that Vidossich had prove that $C_k(X)$ has ccc if $X$ is submetrizable. Who can give a proof of this statement?

Thank you in advance.

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  • $\begingroup$ Don't they give a reference ? $\endgroup$ May 25, 2019 at 5:26
  • $\begingroup$ the given conference is [1972] Vidossich,Function spaces which are pseudo-aleph-compact spaces,preprint $\endgroup$ May 26, 2019 at 1:36
  • $\begingroup$ That' s a bit obscure, indeed. $\endgroup$ May 26, 2019 at 7:21
  • $\begingroup$ Just as an aside: $C_p(X)$ for $X$ Tychonoff (so pointwise topology) is always ccc, being dense in the ccc $\mathbb{R}^X$. $\endgroup$ May 27, 2019 at 5:56

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