Separable quotients in 𝐶_{𝑐}(𝑋), 𝐶_{𝑝}(𝑋), and their duals

Kakol, Jerzy; Saxon, Stephen

doi:10.1090/proc/13360

Separable quotients in $C_{c}( X)$ , $C_{p}( X)$ , and their duals

Authors: Jerzy Kakol and Stephen A. Saxon
Journal: Proc. Amer. Math. Soc. 145 (2017), 3829-3841
MSC (2010): Primary 46A08, 46A30, 54C35
DOI: https://doi.org/10.1090/proc/13360
Published electronically: May 24, 2017
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Abstract: The quotient problem has a positive solution for the weak and strong duals of $C_{c}\left ( X\right )$ ( an infinite Tichonov space), for Banach spaces $C_{c}\left ( X\right )$ , and even for barrelled $C_{c}\left ( X\right )$ , but not for barrelled spaces in general. The solution is unknown for general $C_{c}\left ( X\right )$ . A locally convex space is properly separable if it has a proper dense $\aleph _{0}$ -dimensional subspace. For $C_{c}\left ( X\right )$ quotients, properly separable coincides with infinite-dimensional separable. $C_{c}\left ( X\right )$ has a properly separable algebra quotient if has a compact denumerable set. Relaxing compact to closed, we obtain the converse as well; likewise for $C_{p}\left ( X\right )$ . And the weak dual of $C_{p}\left ( X\right )$ , which always has an $\aleph _{0}$ -dimensional quotient, has no properly separable quotient when is a P-space of a certain special form $X=X_\kappa$

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