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

Ka̧kol, Jerzy; Saxon, Stephen

doi:10.1090/proc/13360

Separable quotients in $C_{c}\left ( X\right )$, $C_{p}\left ( X\right )$, and their duals
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by Jerzy Ka̧kol and Stephen A. Saxon PDF

Proc. Amer. Math. Soc. 145 (2017), 3829-3841 Request permission

Abstract:

The quotient problem has a positive solution for the weak and strong duals of $C_{c}\left ( X\right )$ ($X$ 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 $X$ 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 $X$ is a P-space of a certain special form $X=X_\kappa$

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